Energy Transfer and Dissipation in Nonlinear Oscillators

Energy transfer and dissipation in nonlinear oscillators

Maaita Tzamal- Odysseas

Physicist, MSc. Computational Physics

A thesis submitted for the degree of Doctor of Philosophy

in the

Department of Physics School of Sciences Aristotle University of Thessaloniki

Supervisor: Assistant Professor Efthymia Meletlidou

March, 2014

To my grandmothers Mdallal Enayat and Polidou Marika

Abstract (In Greek)

Στην παρούσα διδακτορική διατριβή μελετάμε ένα σύστημα τριών συζευγμένων ταλαντωτών με τριβή, δύο γραμμικών με έναν μη γραμμικό. Τέτοια συστήματα ταλαντωτών έχουν μεγάλο ενδιαφέρον ιδιαίτερα όταν η μάζα του μη γραμμικού ταλαντωτή είναι πολύ μικρότερη από τους γραμμικούς με συνέπεια ο μη γραμμικός ταλαντωτής να λειτουργεί ως καταβόθρα ενέργειας. Αυτού του είδους τα συστήματα, στα οποία συνυπάρχουν ένας αργός και ένας γρήγορος χρόνος μπορούν να μελετηθούν με τη βοήθεια της singularity analysis, των αναλλοίωτων πολλαπλοτήτων, ενώ σημαντική πληροφορία για τη δυναμική του συστήματος δίνεται και από την δυναμική της αργής ροής (Slow Flow) του συστήματος. Στην παρούσα διατριβή μελετάμε το σύστημα μέσω της μελέτης της αργής αναλλοίωτης πολλαπλότητας (Slow Invariant Manifold- SIM-). Με τη βοήθεια του θεωρήματος του Tikhonov κατηγοριοποιούμε τις διάφορες περιπτώσεις της αργής αναλλοίωτης πολλαπλότητας και ορίζουμε αναλυτικά τις συνθήκες με τις οποίες μπορούμε να οδηγηθούμε στην κάθε περίπτωση. Σε επόμενο βήμα μελετάμε την δυναμική της αργής ροής και παρατηρούμε ότι η δυναμική της είναι πλούσια αφού οι τροχιές της μπορούν να είναι κανονικές, να κάνουν ταλαντώσεις ηρεμίας (relaxation oscillations), ή να είναι χαοτικές. Από την μελέτη της ενέργειας που αποθηκεύεται στον μη γραμμικό ταλαντωτή και από τον ρυθμό απόσβεσης της συνολικής ενέργειας του συστήματος παρατηρούμε ότι τόσο η ύπαρξη διακλαδώσεων της αργής αναλλοίωτης πολλαπλότητας, όσο και η δυναμική της αργής ροής παίζουν καθοριστικό ρόλο στην μεταφορά ενέργειας από τον γραμμικό στον μη γραμμικό ταλαντωτή. Επίσης, στις περιπτώσεις που βλέπουμε μεταφορά ενέργειας παρατηρούμε ότι ο ρυθμός απόσβεσης της συνολικής ενέργειας του συστήματος είναι μεγαλύτερος από τον ρυθμό απόσβεσης όταν δεν μεταφέρεται ενέργεια στον μη γραμμικό ταλαντωτή. Η μελέτη του συστήματος των τριών συζευγμένων ταλαντωτών κλείνει με την πρόταση ενός μη γραμμικού ηλεκτρικου κυκλώματος το οποίο υλοποιεί την μη γραμμική διαφορική εξίσωση δεύτερης τάξης με την οποία προσεγγίσαμε το αρχικό σύστημα. Το συγκεκριμένο κύκλωμα έχει ενδιαφέρον γιατί μας δίνει τη δυνατότητα να μελετήσουμε και πειραματικά διάφορα από τα φαινόμενα που είδαμε στην θεωρητική μας ανάλυση.

Prologue

I was introduced to the subject of this thesis by the late Associate Proffessor of the department of Physics, Simos Ichtiaroglou. Simos Ichtiaroglou participated in the second International Conference on Nonlinear Normal Modes and Localization in Vibrating Systems that took place in Samos at 2006, where he attended a number of paper presentations related to targeted energy transfer. So, when we were discussing the topic of my thesis he proposed to me and to Assistant Professor Efi Meletlidou to work on energy transfer and dissipation because he considered it interesting. The topic was indeed interesting. It included rich theoretical study and practical applications, so we decided to start work on it. The specific system composed of two coupled linear oscillators and a nonlinear oscillator that interacts through an essential nonlinearity with one of the linear oscillators was proposed by proffessor Alex Vakakis in the Euromech Colloquium 503 - Nonlinear Normal Modes, Dimension Reduction and Localization in Vibrating Systems- that took place in Frascati(Rome) at 2009. The work on this thesis was both interesting and challenging. It was interesting because I came into contact with new theories and tools, such as the singular perturbation theory, the multiple scale analysis, the complexification- averaging technique etc... that I was not taught neither in my undergraduate nor graduate studies. It was also interesting because I came into contact with other scientific fields, such as engineering, and I learned a lot of new things, as for example, how we can implement a nonlinear-cubic- term in a mechanical structure. It was challenging, mainly, because there was no financial support and I was obliged to work all the years of my doctoral dissertation outside the university. Our work resulted in four publications in peer reviewed international journals. The first paper is titled “The effect of slow flow dynamics on the oscillations of a singular damped system with an essentially nonlinear attachment” and it was published in the Journal of Applied Nonlinear dynamics in 2013. The second paper is titled “Analytical Homoclinic Solution of a Two-Dimensional Nonlinear System of Differential Equations” and it was published in the

7 Journal of Nonlinear Dynamics in 2013. The third paper is titled “The Study of a Nonlinear Dufﬁng - Type Oscillator Driven by Two Voltage Sources” and it was published in Journal of Engineering Science and Technology Review in 2013. Finaly the fourth paper is titled “The dynamics of the slow ﬂow of a singular damped nonlinear system and its Parametric Study” and it is accepted for publication in the Journal of Applied Nonlinear Dynamics.

This doctorate thesis marks the end of a stage and the begining of a new one in my life. It is therefore time for reasoning. I want to thank my supervisor, Assistant Professor, Efi Meletlidou, for helping me to become a researcher, for teaching me many things, especially on mathematics. She was always understanding and patient and despite the difficulties, we managed to cooperate well. I want to thank my advisor, Associate Professor Iannis Kyprianidis, for his support all these years. His help was important for the implementation of the electric circuit, for the numerical work on computing the maximal Lyapunov Characteristic Exponent. I want to thank my advisor, Professor Alex Vakakis, for the proposal of the system and his guidance on its study. His encouraging words always gave me strength and courage to continue. I want to thank, Dr. Christos Volos, for his help on the inplementation and study of the electric circuit and teaching me the Multisim program. I want to thank, Associate Professor Vassilis Rothos, for his advice on books and papers. Professor George Tsaklidis was a mentor on how to write, check and correct a scientific manuscript. He is a good advisor and a friend. I want to thank, Assistant Professor Menios Tsiganis, for his help and advice, especially, for the Lyapunov Characteristic Exponents and the numerical work. We always have good debates and discussions. I want to thank, Professor Harry Varvoglis, for being there from the beginning of my PhD. Although we disagreed on different issues, he was always there, honest and taking an interest. Assistan Professors George Voyatzis and Harry Skokos, and Special Teaching Fellow Fotini Zervaki, thank you for the support on difficult times. I want to thank, Professor Antonis Anagnostopoulos, for his advice and remarks. Special thanks to my “officemates”, Dr. Stella Tzirti and PhD Candidate Kallinikos Nikos, for the good collaboration and the constructive discussion all these years. I want to thank my friend, PhD Candidate Nikos Bastas, for his help on Grid computing. I want to thank, from the depths of my heart, my parents Ifigeneia and Sami Maaita for all the things that they have done to let me become what i am today. My mother for being by my side, working with me on school, supporting me on difficult times. My father for his guidance, rational thinking and encouragment. Also, my brother Tarek Maaita for always being there when I need him. My wife Kiki Moysiadou for her constant support, for her critical remarks and the continuous

8 interaction. For giving birth to my two beloved little girls Tereza- Leila, Iﬁgeneia- Yasmin and the one that we are waiting to come next June. Finaly, I want to thank all those who contributed in one way or another in the formation of my being. Maaita Jamal- Odysseas, Thessaloniki, Sunday 30th March, 2014.

9 Contents

1 Introduction 15

2 Basic concepts, theorems and tools 21 2.1 Dynamical Systems...... 21 2.1.1 General definitions ...... 21 2.1.2 Equilibrium points (Periodic orbits) and stability: definitions and theorems 22 2.1.3 Bifurcation theory...... 25 2.1.4 Chaos...... 27 2.1.5 The Averaging method...... 29 2.2 Singular Perturbation Theory...... 30 2.2.1 What is a singular perturbation problem? ...... 30 2.2.2 Multiple Scale Analysis...... 30 2.2.3 Geometric Singular Perturbation Theory...... 33 2.3 Electric circuits...... 36 2.3.1 Linear and nonlinear electric elements...... 36 2.3.2 Resistor connections and basic laws ...... 39 2.3.3 Operational Amplifier...... 40 2.3.4 Examples of nonlinear circuits...... 41

3 Classiﬁcation of the SIM 47 3.1 Reduction of the system...... 48 3.2 Singular Perturbation Analysis...... 51 3.3 Solutions and numerical results...... 54 3.4 Conclusions...... 57

4 The dynamics of the slow ﬂow 61 4.1 The dynamics of the slow ﬂow...... 61 4.2 Conclusions...... 71

10 5 Energy transfer and dissipation 77 5.1 The energy of the system...... 77 5.2 Numerical Simulations ...... 78 5.3 Conclusions...... 89

6 Electric circuit 91 6.1 The Nonlinear System and the Proposed Electric Circuit...... 92 6.2 Simulation Results ...... 92 6.3 Conclusions...... 105

7 Epilogue and suggestions for further work 109

A AD734AN 113

B LF411 127

C The homoclinic solution of the hamiltonian part of the slow ﬂow 145

Bibliography 149

11 List of Figures

1.1 The experimental setup which conﬁrmed the phenomenon of Nonlinear Targeted Energy Transfer. a) photo layout. b) Schematic of the two degrees of freedom model (from [48])...... 17

2.1 Saddle equilibrium point: a special case of hyperbolic equilibrium point in systems with two degrees of freedom...... 23 2.2 Branches of equilibrium points...... 26 2.3 Saddle- node bifurcation (Black line: stable equilibrium point, dashed line: unstable equilibrium point)...... 26 2.4 Transciritical bifurcation (Black line: stable equilibrium point, dashed line: unstable equilibrium point)...... 26 2.5 Pitchfork bifurcation (Black line: stable equilibrium point, dashed line: unstable equilibrium point)...... 27 2.6 The oscillation has slow and fast variations...... 30 2.7 Numerical comparison of solution (2.28) (Black dashed line) to the approximate solution(2.32) (gray line) for = 0.01 ...... 32 2.8 Numerical comparison of solution (2.28) (Black dashed line) to the approximate solution(2.40) (gray line) for = 0.01 ...... 33 2.9 Linear Resistor ...... 36 2.10 voltage- current (υ − i) characteristic of a linear resistor...... 36 2.11 Linear capacitor...... 36 2.12 Characteristic of a linear capacitor...... 37 2.13 Linear inductor ...... 37 2.14 Characteristic of a linear inductor...... 37 2.15 Nonlinear elements...... 37 2.16 Ideal diode...... 38 2.17 voltage- current (υ − i) characteristic of an ideal diode ...... 38 2.18 Differential Ampliﬁer...... 40 2.19 Operational Ampliﬁer...... 40

12 2.20 A second order electric circuit with nonlinear capacitor...... 41 2.21 Bifurcation diagrams for the second order electric circuit with nonlinear capacitor . 41 2.22 Trajectories and phase portraits for the second order electric circuit with nonlinear capacitor ...... 42 2.23 A second order electric circuit with nonlinear inductor [38]...... 43 2.24 Bifurcation diagram for the second order electric circuit with nonlinear inductor . . 43 2.25 A second order electric circuit with nonlinear resistor...... 44 2.26 Bifurcation diagram for the second order electric circuit with nonlinear resistor . . 44 2.27 Poincare sections for the Dufﬁng- van der Pol electric circuit with nonlinear Resistor 44 2.28 Electric circuit with nonlinear resistor of type-N...... 45 2.29 Characteristic of the nonlinear resistor...... 45 2.30 Experimental implementation of a type N nonlinear resistor...... 46 2.31 Bifurcation diagrams of the electric circuit with a nonlinear resistor of type-N . . . 46

3.1 The polynomial P (z) for different values of λˆ ...... 54 3.2 The case where the SIM has always one stable branch (gray solid line: SIM, black dashed line: N(t) of (3.16))...... 55 3.3 The case where the SIM has always three branches (gray solid line: SIM, black dashed line: N(t) of (3.16))...... 56 3.4 Relaxation oscillations (gray solid line: SIM, black dashed line: N(t) of (3.16)) . . 57 3.5 Relaxation Oscillations of the slow ﬂow when the SIM has the structure of case 4, A˜ = −0.093, Bˆ = −0.175,J = 0.105, Cˆ = 2, = 0.1, λˆ = 0.09 ...... 58 3.6 Bifurcations between the upper (or the lower) stable branch and the unstable branch of the SIM (gray solid line: SIM, black dashed line: N(t) of (3.16)) ...... 58 3.7 Excitation of the orbits (gray solid line: SIM, black dashed line: N(t) of (3.16)) . . 59

4.1 A˜ = −0.075, Bˆ = 0.614,J = 0.063, = 0.1, B¯ = −0.138 ...... 64 ω20 4.2 Regular orbits...... 65 4.3 A˜ = −0.113, Bˆ = 0.124,J = 0.135, = 0.1, B¯ = −0.073 ...... 66 ω20 4.4 bifurcation diagram for A˜ = −0.113, Bˆ = 0.206,J = 0.135, = 0.1, B¯ = −0.073 67 ω20 4.5 A˜ = −0.113, Bˆ = 0.206,J = 0.135, = 0.1, B¯ = −0.073 ...... 68 ω20 4.6 bifurcation diagram for A˜ = −0.093, Bˆ = −0.175,J = 0.105, = 0.1, B¯ = −0.055 69 ω20 4.7 A˜ = −0.093, Bˆ = −0.175,J = 0.105, = 0.1, B¯ = −0.055 ...... 69 ω20 4.8 Examples of relaxation oscillations...... 70 4.9 bifurcation diagram for A˜ = −0.111, Bˆ = 0.178,J = 0.109, = 0.1, B¯ = −0.335 72 ω20 4.10 A˜ = −0.111, Bˆ = 0.178,J = 0.109, = 0.1, B¯ = −0.335, λˆ = 0.01 ...... 73 ω20 4.11 bifurcation diagram for A˜ = −0.127, Bˆ = 0.139,J = 0.125, = 0.1, B¯ = −0.328 74 ω20 4.12 A˜ = −0.127, Bˆ = 0.139,J = 0.125, = 0.1, B¯ = −0.328, λˆ = 0.059 ...... 75 ω20 ˆ 5.1 One stable branch of the SIM: a = 0.1, d = 1.0, λ = 0.8, x0 = 0.09, x1 = 0.05 . . . 79

13 ˆ 5.2 One stable branch of the SIM: a = 0.1, d = 0.5, λ = 0.3, x0 = 0.01, x1 = 0.01 . . . 80 ˆ 5.3 One stable branch of the SIM: a = 0.2, d = 0.2, λ = 0.5, x0 = 0.01, x1 = 0.01 . . . 81 ˆ 5.4 One stable branch of the SIM: a = 0.01, d = 1.0, λ = 0.1, x0 = 0.9, x1 = 0.5 . . . 82 ˆ 5.5 One stable branch of the SIM: a = 0.2, d = 0.2, λ = 0.05, x0 = 0.01, x1 = 0.01 . . 83 ˆ 5.6 Always three branches of the SIM: a = 0.01, d = 1.5, λ = 0.1, x0 = 0.09, x1 = 0.05 84 ˆ 5.7 Relaxation oscillations of the slow ﬂow: a = 4, d = 4, λ = 0.15, x0 = 0.9, x1 = 0.5 85 ˆ 5.8 The SIM bifurcates: a = 6, d = 6, λ = 0.5, x0 = 0.9, x1 = 0.04 ...... 86 ˆ 5.9 The SIM bifurcates: a = 6, d = 6, λ = 0.15, x0 = 0.9, x1 = 0.5 ...... 87 5.10 Oscillations of the initial system when the SIM has no bifurcations. Black line: y(t),

thick gray line: x0(t), dashed gray line: x1(t) ...... 88 5.11 Oscillations of the initial system when the SIM bifurcates. Black line: y(t), thick

gray line: x0(t), dashed gray line: x1(t) ...... 89

6.1 The proposed circuit emulating the nonlinear system...... 93

6.2 Bifurcation diagrams x vs. U0, for λ = 0.1...... 94

6.3 Phase portraits for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 4 and B = 0. 95

6.4 Phase portraits for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 0 and B = 4. 96

6.5 Phase portraits for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 4 and B = 4. 97

6.6 Poincare section for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 4 and B = 4. Multiband chaos ...... 98

6.7 Phase portraits for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 7 and B = 15. 99

6.8 Poincare section for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 7 and B = 15. Strange Attractors...... 100

6.9 Bifurcation diagrams, x vs. U0, when only one voltage source is activated . . . . . 101 6.10 For higher values of the damping factor λ, or for higher values of the frequency, period-1 oscillations are observed, as the value of the amplitude of the voltage source

is increased. Bifurcation diagrams x vs. U0...... 102

6.11 Phase portraits for λ = 0.1, f11 = 0.7 and f22 = 1.835, A = −20.2, B = 82.3 and C = 0.2...... 103

6.12 Phase portraits of y vs.x for λ = 0.1, f11 = 0.7 and f22 = 1.835, A = 20.2, B = 82.3 and C = 0.2...... 104

6.13 Poincare section for λ = 0.1, f11 = 0.7 and f22 = 1.835 in the case A = 20.2 and B = 82.3 and C = 0.2. A case of quasiperiodicity...... 105

6.14 Phase portraits of y vs.x for λ = 0.15, f11 = 0.7 and f22 = 1.835, A = 20.2, B = 82.3 and C = 0.2...... 106

6.15 Poincare section for λ = 0.15, f11 = 0.7 and f22 = 1.835 in the case A = 20.2 and B = 82.3 and C = 0.2. A paradigm of 3-tori quasiperiodicity...... 107

π C.1 phase space of the hamiltonian for A = 0.1,B = 0.001,J = 0.00001, γ = 9 and C = 2.0...... 148

14 Chapter 1

Introduction

Oscillations are seen in many phenomena in nature and used in many mechanical, electrical, biological and other applications. There are many studies concerning oscillations. In our work we focused in studying a singular system of linear and nonlinear oscillators because in such systems the phenomenon of energy transfer occurs. The one way directed energy transfer from an initial system (donor) to a final system (receptor) is called Targeted Energy Transfer [63] and it plays an important role in many physical phenomena. In biology, it plays a role in the process of energy transfer during the photosynthesis, that is, the energy from the Sun is captured by photobiological antenna chromophores and is then transfered to reaction centers through a series of interactions between chromofore units. In dynamics, on systems of nonlinear oscillators where the energy transfers from linear to nonlinear oscillators. In materials, on applications of superconductivity etc. An important application of the Targeted Energy Transfer is the development of reliable and efficient energy harvesting devices, which can collect energy from various external sources and make use of it. Such examples have been studied on mechanical systems that convert external oscillations into useful electrical energy [63]. The main mechanism behind the Targeted Energy Transfer is the resonances between the donor and the receptor [54, 55]. Particularly, in the case of a single mode energy attractors, the energy transfer occurs in a restricted band of frequencies that are related with the given frequency of the system. In the present thesis we will deal with mechanical systems connected to a local attachment possessing essential stiffness nonlinearity (commonly, purely cubic spring). The addition of such nonlinear attachments to the linear systems significantly effect the overall dynamics of the system. The basic reason of this impact is the lack of preferential resonance frequency of the nonlinear attachment. This enables the nonlinear attachment to engage in nonlinear resonance with any mode of the linear system, giving the ability of resonances over a broad frequency range.

15 This property of the local attachment with essential stiffness nonlinearity gives the ability to use such attachments as Nonlinear Energy Sinks (NESs). Indeed, under certain initial conditions a one-way, irreversible (on the average) flow of energy from the linear to the nonlinear attachment occurs. This phenomenon is called nonlinear Targeted Energy Transfer. The Nonlinear Energy Sink may participate in a set of transitive nonlinear resonances. Depending on the instantaneous energy level the Nonlinear Energy Sink transitionally tunes, for a finite time, and exports energy. Then “escapes” from this resonance because of energy dissipation and participates in another transitional tune with a different frequency level. The control parameter of this procedure is the instantaneous energy level of the system. The Nonlinear Energy Sink can be adjusted in a way that it will firstly extract energy from rich energy states and then tuned in a way to extract energy from states with less energy. This makes it particularly suitable in practical applications, where it is necessary to extract energy from multiple ways of oscillation. The cause of this behavior of the Nonlinear Energy Sink lies in the fact that the addition of an essential nonlinear attachment to the linear system introduces high degeneracy in the dynamics of the system and gives the ability for higher co-dimensional bifurcations and complex dynamical phenomena. Gendelman [14] was the first to observe and study the nonlinear Targeted Energy Transfer. He studied the transient dynamics of a two degrees of freedom system consisting of a linear oscillator with damping coupled weakly with an essential nonlinear oscillator. On this system, he observed that for certain initial conditions of energy given in the linear oscillator (above a threshold) there was a Targeted Energy Transfer to the nonlinear oscillator, which absorbed most of the energy. More extensive study of the above phenomenon [12, 65] proved that the basic mechanism that controls Targeted Energy Transfer was the 1:1 transitional resonance between the linear and the nonlinear system, that is the 1:1 resonance occurs and continues for a certain period of time, followed by a transition to escape from resonance. Also it was proved that the existence of dissipation is a prerequisite for Targeted Energy Transfer. Whereas, in the absence of dissipation beating phenomena occurs, i.e. the energy is transferred from the linear to the nonlinear and back again to the linear oscillator. The first experimental demonstration of a Targeted Energy Transfer in a mechanical system took place in 2005 by McFarland, Bergman and Vakakis [48], where they studied a system consisting of two, one degree of freedom, oscillators (one linear and one nonlinear) with a stiff linear coupling and a dissipative Nonlinear Energy Sink (Figure 1.1). With the above experiment, the researchers noticed good agreement between theoretical and experimental results, despite the strong nonlinearity and the transient nature of the dynamics of the problem. Indeed, the energy which was introduced in the linear oscillator was transferred to the Nonlinear Energy Sink where it was locally dissipated. The results of the above experiment confirmed the existence of an energy threshold after which apearss the phenomenon of Targeted Energy Transfer. The initial energy of the system ( related to the amplitude of the oscillation) plays an essential

16 Figure 1.1: The experimental setup which conﬁrmed the phenomenon of Nonlinear Targeted Energy Transfer. a) photo layout. b) Schematic of the two degrees of freedom model (from [48]).

role on the Targeted Energy Transfer phenomenon. This fact is related to the mechanisms of energy transfer. Certain criteria that relates the initial energy (amplitude) with the Targeted Energy Transfer were defined in different papers [64]. The phenomenon of Targeted Energy Transfer may be interpreted by three mechanisms. The first mechanism is the basic (1:1) resonance between the donor and the receptor that leads to the basic Targeted Energy Transfer. The second mechanism is the subharmonic resonance that leads to the subharmonic Targeted Energy Transfer. The third mechanism is based on the excitation of the so-called impulsive periodic or semi-periodic orbits which leads through rhythmic phenomena to one of the previous two mechanisms. Singular Perturbation Theory and Invariant Manifolds are important mathematical tools [18, 19, 69] that help us study systems with Targeted Energy Transfer. This is due to the fact that the mass of the Nonlinear Energy Sink is much smaller in comparison to the linear system and the system’s dynamics are governed by different time scales. The first who studied such systems was Gendelman [13], who showed that the rate of dissipation of the energy is connected to the bifurcations of the invariant manifold.

17 The slow flow dynamics of the system is the essential on the Targeted Energy Transfer phenomenon [36, 53, 63]. Therefore techniques as the Complexification Averaging Technique (CX-A) are used to separate and study the slow flow dynamics of the system. Targeted Energy Transfer occurs even in the case where the Nonlinear Energy Sink is attached to a system with external excitations. In this case the system, in addition to the known steady state oscillations with the weak variation of the responses, perform very special oscillations with large variations of the responses which are called Strongly Modulated Responses. These responses can be considered as the extension of the phenomenon of Targeted Energy Transfer in systems with external excitations since the Strongly Modulated Responses can be considered as a form of iterative Targeted Energy Transfer. The applications of the Targeted Energy Transfer have great interest. In recent years, intense efforts have been done in this area. Papers have been published concerning seismic mitigation of energy in buildings and the use of the Targeted Energy Transfer phenomenon to transfer and dissipate the energy in order to protect the buildings [51, 60]. There are studies on the application of the Targeted Energy Transfer to stabilize drill-string systems [71] and studies on transferring of unused energy in automotive gearboxes systems [35]. In the following thesis we study a three degree of freedom autonomous system with damping, composed of two linear coupled oscillators with an essentially nonlinear lightweight attachment. A similar work was done by Vakakis [66]. He studied the dynamic interactions between traveling waves propagating in a linear lattice and a lightweight, essentially nonlinear and damped local attachment through slow-fast partitions of the dynamics and Melnikov analysis. He proved that for damping below a critical threshold relaxation oscillations of the attachment exist. These oscillations are associated with enhanced targeted energy transfer from the traveling wave to the attachment. Moreover, in the limit of weak or no damping, he proved the existence of subharmonic oscillations of arbitrarily large periods, and of chaotic motions. In our work we study, with the help of the Slow Invariant Manifold (SIM) approach, how the parameters of the original problem influence the asymptotic behavior of the orbits of the system. This is accomplished with the application of Tikhonov’s theorem [69]. We classify the different cases of the dynamics according to the values of the parameters and the theoretically predicted asymptotic behavior of the orbits. In the next step we study the dynamical behavior of the slow flow of the system. The dynamics of the slow flow can be either simple (having regular oscillations in the region of the stable branches of the SIM), having relaxation oscillations or chaotic behavior. The study of the slow flow and the bifurcations of the SIM are related with the Targeted Energy Transfer of the initial problem. The classification of the different cases of the SIM gives us, in a way, a way to control the dynamics of the system. Furthermore, an electric circuit that implements the nonlinear reduced system is presented. This electric circuit can be used for the experimental study as well as for developing applications that use the rich dynamics of the above system.

18 This thesis is organized as follows. In the next chapter we present some basic concepts, theorems and tools that we use in our study. The third chapter presents the bifurcations of the SIM and a classification for the different cases. In the fourth chapter we present the dynamics of the slow flow. The effect of the SIM and the slow flow dynamics on the energy transfer and dissipation of the initial system is presented in the fifth chapter. The sixth chapter presents an electrical circuit that implements the reduced nonlinear system. Finally, in the seventh chapter, we present the conclusions of our work and suggestions for further work.

Chapter 2

Basic concepts, theorems and tools

In what follows we present basic concepts, theorems and tools that are used in our study. In the ﬁrst section we present general deﬁnitions and basic theorems concerning dynamical systems and chaos. In the second section we make a small introduction on singular perturbation theory, presenting the multiple scale analysis and basic theorems of the geometric singular perturbation theory. Finally, the chapter ends with a small introduction about nonlinear electric elements.

2.1 Dynamical Systems

2.1.1 General deﬁnitions

Systems that evolve depending on time are called Dynamical Systems[1, 10, 31, 70]. Such systems can be classiﬁed into two categories: continuous dynamical systems, where the time is a continuous parameter (t ∈ <) and are mathematically described by differential equations of the form dx i =x ˙ = f (x , t), i, j = 1, ..., n, (2.1) dt i j i and discrete dynamical systems, where time evolves in a discrete way (t ∈ Z or N) and are mathematically described by iterated maps

k+1 k xi = fj(xi ), i, j = 1, ..., n, k = 1, 2, .... (2.2)

In the current work we will study continuous dynamical systems. The systems that do not explicitly depend on time are called autonomous dynamical systems. Such systems are given by

x˙i = fj(xi), i, j = 1, ..., n. (2.3) The systems that depend explicitly on time are called non autonomous dynamical systems

x˙i = fj(xi, t), i, j = 1, ..., n. (2.4)

21 The number n of the variables x deﬁnes the dimension of the system. The (n+1)-dimensional space (x1, x2, ..., xn, t) is called the extended state space. For autonomous systems we can use the n-dimensional space of the positions (x1, , x2, ..., xn) that is called state space. According to the theorem of Existence and Uniqueness of the solutions, the trajectories in the extended state space cannot intersect. On the other hand the trajectories in the state space can not intersect unless fj(xi) = 0.

In a continuous dynamical system, the parametric solution (x1(t), x2(t), ..., xn(t)) can be represented as a continuous curve in the state space that is named trajectory. Respectively in a discrete dynamical system, the trajectory of the system is the countable set of the points (k) (k) (k) (x1 , , x2 , ..., xn ), where k = 1, 2, ..., +∞. If the system is invertible then the trajectories can (k) (k) (k) be deﬁned as the countable set of points (x1 , x2 , ..., xn ), where k = −∞, ..., +∞. Periodic orbit is the orbit that returns to the same point after one period of time (T ), i.e, for t = T , holds xi(t + T ; xj0, t0) = xi(t, xj0, t0).

2.1.2 Equilibrium points (Periodic orbits) and stability: deﬁnitions and theorems

Equilibrium points are the critical points where the state of the dynamical system remains static in dependence to time,

x˙i = 0. (2.5) The stability of the equilibrium points is very important because it determines the dynamics of the system [22]-[25]. In particular, a stable equilibrium point is a point for which the trajectories around it remain close for small perturbations. On the other hand, an unstable equilibrium point is a point for which the trajectories around it escape even for small perturbations and remove the system from its initial state. The stability of the equilibrium points can be studied by linearizing the system in the region of the equilibrium point.

Let x0 be an equilibrium point of the nonlinear system (2.3). We consider the perturbed solution x(t), that is the displacement from the equilibrium point and satisfy system (2.3), replace it in (2.3) and expand the right hand side in Taylor Series

x˙ = f(x0) + Df(x0)x + ..., x ∈ <. (2.6)

By keeping O(1) terms and taking into consideration that in the equilibrium point f(x0) = 0 we derive the linear system x˙ = Ax, (2.7) where the matrix A is

 ∂f1 ... ∂f1  ∂x1 ∂xn   A = Df(x0) =  ......  . (2.8) ∂fn ... ∂fn ∂x1 ∂xn x=x0

22 Respectively, for the periodic orbits, a neighboring orbit to the periodic, x(t), is given by x(t) = γ(t) + ξ(t) and the linearized system is

ξ˙ = Aξ, (2.9) where the matrix  ∂f1 ... ∂f1  ∂x1 ∂xn   A = Df(γ(t)) =  ......  , (2.10) ∂fn ... ∂fn ∂x1 ∂xn x=γ(t) is periodic with period T .

In general, when the eigenvalues of matrix A of an equilibrium point p0 have non zero real part, this equilibrium point is called hyperbolic equilibrium point. The solution of the linearized system determines the behavior of the nonlinear system in the region of equilibrium point (or periodic orbit) in a linear approach. The stability of the system is determined by the eigenvalues of the linearized matrix A of the equilibrium point. In particular, if the matrix A has at least one eigenvalue with positive real part then the equilibrium point is unstable. If all the eigenvalues of A have negative real parts then the equilibrium point is stable.

Figure 2.1: Saddle equilibrium point: a special case of hyperbolic equilibrium point in systems with two degrees of freedom.

Topological conjugacy allows us to study a nonlinear system through linearization. Two systems of ﬁrst order differential equations, φt(x), ψt(x), are called topologically conjugated when they are topologically equivalent, i.e., when there is a continuous function h which is reversible by a continuous inversion and

φt ◦ h(x) = h ◦ ψt(x) ⇔ φt(h(x)) = h(ψt(x)), and maintains the parametrization of the solutions with respect to time. The topological conjugacy shows that the orbits of φt(x) are mapped in the orbits of ψt(x) and therefore both systems have the same topological properties.

23 We deﬁne the locally stable manifold S, as the manifold that has the property

S = {x ∈ S/φt(x) → x0, t → ∞ & φt(x) ∈ S, t ≥ 0}, (2.11) and the locally unstable manifold U, as the manifold that has the property

U = {x ∈ U/φt(x) → x0, t → −∞ & φt(x) ∈ S, t ≤ 0}. (2.12)

We deﬁne the unstable subspace Eu = span{u1, ..., unu }, where u1, ..., unu are the eigenvectors of (2.10) whose eigenvalues have negative real parts, as the set of points such that orbits of (2.9), through these points, approach the origin asymptotically in negative time. We deﬁne the stable subspace Es = span{u1, ..., uns }, where u1, ..., uns are the eigenvectors of (2.10) whose eigenvalues have negative real part, as the set of points that the orbits of (2.9), through these points, approach the origin asymptotically in positive time. The global stable manifold is given by

s [ W = φt(S), (2.13) t≤0 and the global unstable manifold by

u [ W = φt(U). (2.14) t≥0 In what follows we present basic theorems that connect the linearized with the nonlinear system [77].

Theorems

Stable manifold theorem: Let the dynamical system x˙ = f(x), x ∈

Hartman - Grobman theorem: In an open subset E ∈

24 The system orbits x˙ = f(x) are mapped to orbits of the linearized system x˙ = Ax in the region of the zero point and maintain the parameterization. Based on the above, we conclude that the stable and the unstable subspace in the region of the equilibrium point, as well as the topology of the phase space, remain the same when the matrix A of the linearized system does not have eigenvalues with zero real part.

2.1.3 Bifurcation theory

If the topology of the vector ﬁeld f of the dynamical system

x˙ = f(x, µ), µ ∈ <, f ∈ C1(E),E ⊂

Continuation theorem: Let the system

x˙ = f(x, µ), x ∈

Jacobian matrix has an eigenvalue with zero real part the point (x0, µ0) is called bifurcation point. From a bifurcation point, more than one branch of equilibrium points may pass (Figure 2.2).

Types of bifurcations

1. Saddle - node bifurcation: In this case, for the critical value of the parameter µ, for which the eigenvalues of the linearized matrix in the region of the equilibrium point are real and equal to zero, the two equilibrium points suddenly appear or if they exist one coalesce with the other and suddenly disappear (Figure 2.3).

2. Transciritical bifurcation: In this case, there are always two equilibrium points, one stable and one unstable, that for the value of the parameter µ for which the two equilibrium points coincide, switch their stability status, i.e. the unstable becomes stable and the stable unstable (Figure 2.4).

25 Figure 2.2: Branches of equilibrium points

Figure 2.3: Saddle- node bifurcation (Black line: stable equilibrium point, dashed line: unstable equilibrium point).

Figure 2.4: Transciritical bifurcation (Black line: stable equilibrium point, dashed line: unstable equilibrium point).

26 3. Pitchfork bifurcation: Here, there is one equilibrium point that, for the critical value of the parameter µ, changes its stability and simultaneously two other equilibrium points appear with the same stability as the original (Figure 2.5).

Figure 2.5: Pitchfork bifurcation (Black line: stable equilibrium point, dashed line: unstable equilibrium point).

4. Hopf bifurcation: In this bifurcation, one branch of periodical orbits bifurcates from one branch of equilibrium points. This behavior is related to a single pair of imaginary eigenvalues (real parts equal to zero) and no other eigenvalue with zero real part.

2.1.4 Chaos

A dynamical system is called chaotic when it has a sensitive dependence on the initial conditions [2,3,4, 29, 30]. In particular, a map 1 f is called chaotic in a compact invariant set when:

• It has a dense set of periodic orbits.

• It is topologically transitive. This means that from any arbitrarily small region containing points of the invariant set, we may visit any other region that contains points of the invariant set as well. This property results in a mixing of the orbits under the mapping.

• It has a sensitive dependence on initial conditions. This means that in every region of every point Π of the set, there is at least one point Π0 such, that the orbits of the two points diverge a given ﬁxed distance, independent of the starting point and the selected region, after ﬁnite steps.

The detection of chaotic behavior is a challenging process. There are different analytic, semi-analytic or numerical tools to investigate the behavior of the system, such as: Melnikov’s

1The maps f : <2 → <2 has similar properties with the Poincare maps in the phase space of three dimensional time continuous dynamical system.

27 Theorem, Lyapunov Characteristic Exponent, SALI,... etc [4, 56, 77]. These tools may prove or show the existence of chaotic behavior in a dynamical system.

Melnikov theorem

The Melnikov theorem is one of the few analytical tools that can guarantee the chaotic behavior in a dynamical system [77]. Consider the dynamical system of the form

x˙ = f(x) + g(x, t) (2.17)

2 r where x ≡ (x1, x2) ∈ < , f ≡ (f1, f2), g ≡ (g1, g2) are C (r ≥ 2) smooth functions, gi(x, t) = gi(x, t + T ) are periodic functions of time and ∈ < a small parameter of the system.

The unperturbed system must have one hyperbolic (saddle) equilibrium point p0 for which u s 2 the unstable W (p0) and the stable manifold W (p0) join in a smooth homoclinic trajectory . Also there is a continuous inﬁnity of periodic orbit inside the homoclinic trajectory.The Melnikov function (or integral) is given by Z ∞ M(t0) = f(q0(t)) ∧ g(q0(t), t + t0)dt. (2.18) −∞

Melnikov theorem: If the Melnikov function M(t0) has simple roots then, for a small enough ,

u t0 s t0 the invariant manifolds W (p ) and W (p ) intersect transversely. These manifolds are the two dimensional, unstable and stable respectively, manifolds of the periodic orbit that the perturbed system has and is continued from the saddle equilibrium point of the unperturbed system. The existence of the transverse homoclinic trajectories implies, through the Smale- Birkhoff theorem, that the Poincare mapping has an invariant hyperbolic set, smale horseshoe and chaotic dynamics. This means that the above theorem proves the existence, for small values of the perturbation parameter , of the chaotic behavior of the system.

Maximal Lyapunov Characteristic Exponent (mLCE)

The Lyapunov Characteristic Exponent is an asymptotic measure that gives the average increase (or decrease) of the deviation of the perturbed initial conditions in the solutions of a dynamical system. The maximal Lyapunov Characteristic Exponent is an indicator of the nature of the motion (chaotic or regular) [56, 61]. In order to have chaotic motion, the maximal Lyapunov Exponent must be positive. The maximal Lyapunov Characteristic Exponent can be computed by

1 kw(t)k χ = lim . (2.19) 1 t→∞ t kw(0)k 2Homoclinic is called the trajectory of a point p ∈ <2 when it tends asymptotically, for n → ∞ and n → −∞, to an equilibrium point or a periodic orbit or generally to any invariant set.

28 where w is the divergence vector and is calculated from the solutions

 ∂f ∂f ∂f  1 1 ··· 1 ∂x1 ∂x2 ∂xn  ∂f2 ∂f2 ··· ∂f2   ∂x1 ∂x2 ∂xn  w˙ (t) =  . . .  w(t). (2.20)  ......   . . .  ∂fn ∂fn ··· ∂fn ∂x1 ∂x2 ∂xn The maximal Lyapunov Characteristic Exponent gives also an estimate about the time that the system needs to become chaotic. This time is called Lyapunov time and is given by 1 tL = . (2.21) χ1

2.1.5 The Averaging method

The Averaging method gives important information on the evolution of a nonlinear system [26]. This method can be applied in systems of the form

x˙ = f(x, t, ), x ∈

1 Z T y˙ = f(y, t, 0)dt ≡ f¯(t). (2.23) T 0

The Averaging theorem [20]: There is a Cr transformation of the coordinates x = y+w(y, t, ) such that (2.22) becomes ¯ 2 y˙ = f(y) + f1(y, t, ), (2.24) where f1 is periodic with period T . Also

• If x(t), y(t) are the solutions of (2.22) and (2.23) respectively for the initial conditions

x0, y0, for time t = 0, and |x0 − y0| = O(), then the relation |x(t) − y(t)| = O() holds 1 for time t ∼ .

• If p0 is a hyperbolic equilibrium point of (2.23) then there is > 0 such that, for every

0 < ≤ 0,(2.22) has one unique hyperbolic periodic orbit γ(t) = p0 + O() that has the

same stability with the equilibrium point p0.

s s • If x (t) ∈ W (γ) is the solution of (2.22) on the stable manifold of the hyperbolic s s periodic orbits γ = p0 + O(), and y (t) ∈ W (p0) the solution of (2.23) on the stable s s manifold of the hyperbolic equilibrium point p0 and holds that |x (0) − y (0)| = O(), then |xs(t) − ys(t)| = O() for every t ∈ [0, ∞). Similar results hold for solutions that are on the unstable manifolds for time t ∈ (−∞, 0].

29 2.2 Singular Perturbation Theory

2.2.1 What is a singular perturbation problem?

Many physical problems involve small, real valued parameters. Examples of such systems are seen in many different scientific fields such as celestial mechanics, electric circuits, biology systems or chemical reactions. The study of this kind of problems can be done with the use of perturbation theory. The aim of perturbation theory is to determine the behavior of the solutions when the value of the small parameter tends to zero, → 0. This is done with the use of asymptotic solutions that satisfies the original system up to a small error. In regular perturbation problem the expansion with respect to a small parameter depends analytically on it. Singular perturbation problems are qualitatively different from regular perturbation problems. The expansion of the solution depends singularly on the small parameter . The solutions of singular perturbation problems often depend on widely different time-scale lengths. There are many methods to tackle this kind of problems such as multiple scale analysis and averaging. The main idea in these methods is to separate time to fast and slow time variables and treat them as if they are independent.

2.2.2 Multiple Scale Analysis

Multiple Scale Analysis are techniques that give approximate solutions to singular perturbation problems, where the dynamics display slow and fast variations (Figure 2.6)[28, 37, 49].

Figure 2.6: The oscillation has slow and fast variations

The basic idea of Multiple Scale Analysis is to separate an independent variable, for example the time t, into other independent, scaled variables. i.e, to separate the time variable t into 2 variables t0 = t, t1 = t, t2 = t, .... that are independent to each other.

30 These techniques were developed in the early decade of 1960 with the work of Kuzmak (1959), Cochran (1962), Mahony (1962) and Nayfeh (1964). Since then Multiple Scale Analysis techniques were, and still are, used in many ﬁelds of physical sciences (Ordinary differential equations, Partial differential equations, problems of space science, aeronautic, solid mechanics, waves, atmospheric physics, plasma ...etc).

General methodology

Consider the differential equations of the form dnx(t) = f(x(t)). (2.25) dtn

We introduce the new time variables t0 = t, t1 = t, ... and expand the solution x(t) in a series

x(t) = X0(t0, t1) + X1(t0, t1) + .... (2.26) With the help of the chain rule we ﬁnd the derivatives of the solution dx ∂X dt ∂X dt ∂X dt ∂X dt = 0 0 + 0 1 + ( 1 0 + 1 1 ) + ..., dt ∂t0 dt ∂t1 dt ∂t0 dt ∂t1 dt ∂X ∂X ∂X = 0 + ( 0 + 1 ) + O(2), ∂t ∂t1 ∂t 2 2 2 d x ∂ X0 ∂X0 ∂ X1 2 2 = 2 + (2 + 2 ) + O( ), dt ∂t ∂t∂t1 ∂t ··· . (2.27)

By substituting (2.27) and (2.26) in the differential equation and after equating terms of the same order we lead to a set of subproblems. The solutions of these subproblems will give the solution of the initial problem.

An illustrative example

Consider the Dufﬁng Equation with no damping and no excitation d2y + y + y3 = 0 (2.28) dt2 dy with initial conditions y(0) = 1, dt = 0. The ﬁrst step in order to study the problem is to take the expansion of the solution y(t) in a series 2 y(t) = y0 + y1 + O( ). (2.29) The second step is to substitute the solution (2.29) in equation (2.28), and by equating terms of the same order we take the subproblems

y¨0 + y0 = 0, 3 y¨1 + y1 + y0 = 0. (2.30)

31 The solutions of (2.30) are

y0 = cos(t), 1 3 y = (cos(3t) − cos(t)) − t sin(t) (2.31) 1 32 8 and the solution of the initial problem is 1 3 y(t) = cos(t) + ( (cos(3t) − cos(t)) − t sin(t)) + O(2). (2.32) 32 8 3 It may be observed in (2.32) that the term 8 t sin(t) is an increasing function of time t. It can be seen in (Figure 2.7) that our approach is quite good only for a short time period, while with the increasing of time the solutions diverge. The time for which there is good approximation depends on .

(a) t ∈ (0, 200) (b) t ∈ (0, 800) Figure 2.7: Numerical comparison of solution (2.28) (Black dashed line) to the approximate solution(2.32) (gray line) for = 0.01

The above difﬁculty can be avoided by introducing the method of multiple scale analysis.

The ﬁrst step is to consider the fast time t0 = t and the slow time t1 = t and expand the solution y(t) of (2.28) as in (2.26). The solution becomes

2 y(t) = Y0(t0, t1) + Y1(t0, t1) + O( ) (2.33) with derivatives dy ∂Y ∂Y ∂Y = 0 + ( 0 + 1 ) + O(2), dt ∂t ∂t1 ∂t 2 2 2 d y ∂ Y0 ∂Y0 ∂ Y1 2 2 = 2 + (2 + 2 ) + O( ). (2.34) dt ∂t ∂t∂t1 ∂t By substituting the solution (2.33) and the derivatives (2.34) in (2.28) and by equating terms of the same order we derive the subproblems ∂2Y 0 + Y = 0, ∂t2 0 2 2 ∂ Y1 3 ∂ Y0 2 + Y1 = −Y0 − 2 . (2.35) ∂t ∂t∂t1

32 (a) t ∈ (0, 200) (b) t ∈ (0, 800) Figure 2.8: Numerical comparison of solution (2.28) (Black dashed line) to the approximate solution(2.40) (gray line) for = 0.01

The ﬁrst equation of(2.35) has a general solution

jt ∗ −jt Y0(t0, t1) = A(t1)e + A (t1)e , (2.36) where A(t1) is complex. By substituting on the right hand-side of the second subproblem (2.35) we take ∂2Y dA 1 + Y = −3A3e3jt + (−3A2A∗ − 2j )ejt ∂t2 1 dt dA∗ − 3(A∗)3e−3jt + (−3A(A2)∗ − 2j )e−jt. (2.37) dt The occurrence of secular terms can be prevented by imposing the condition dA − 3A2A∗ − 2j = 0, (2.38) dt with solution 1 3 jt A = e 8 1 . (2.39) 2 Then the approximate solution of the Dufﬁng equation is 3 y(t) = cos((1 + )t) + O(). (2.40) 8 Comparing the numerical result, given by solving the differential equation (2.28), and the above approximate solution we see that there is a good approximation of the numerically calculated solution by equation (2.40) (Figure 2.8) and conclude that the multiple scale analysis is a good method to approximate of the solutions of the above problem.

2.2.3 Geometric Singular Perturbation Theory

A powerful tool for analyzing high-dimensional singular systems is the Geometric Singular Perturbation Theory [27, 28, 33, 77]. A map f : X → Y of subsets of two Euclidean spaces is called Cr diffeomorphism if it is one to one and onto and if the inverse map f −1 : Y → X is also Cr. A subset M ⊂

33 • There exists a countable collection of open sets V a ⊂

• There exist a Cr diffeomorphism xa deﬁned on each U a which maps U a onto some open set in

• The change of the coordinates in the overlapping region of two of the open sets U a must also be Cr.

A manifold contained in the phase space of a dynamical system which has the property that its orbits remain on the manifold throughout the course of their dynamical evolution is called Invariant Manifold. We consider the equations of the form

dx = f(x, y, ), dt dy = g(x, y, ), (2.41) dt where x ∈

dx = f(x, y, ), dτ dy = g(x, y, ). (2.42) dτ System (2.41) is called fast and (2.42) is called slow system.

Let assume an l-dimensional manifold, possibly with its boundary, M0 which is contained in the set {f(x, y, 0) = 0} and the fundamental hypothesis on M0 that as a set of critical points, the directions normal to the manifold will correspond to eigenvalues that are not neutral. A set M is locally invariant under the ﬂow form (2.41) if it has a neighborhood V so that the trajectory can not leave M without also leaving V .

In what follows we present a set of theorems that play important role in the geometric singular perturbation theorem [33, 69, 77].

Theorems

Fenichel’s Invariant Manifold Theorem I: If > 0 but sufﬁciently small, there exist a manifold M that lies within O() of M0 and is diffeomorphic to M0. Moreover it is locally invariant under the ﬂow of system (2.41), and Cr, including in , for any r < +∞.

The manifold M is called the slow manifold and it is locally invariant.

34 Fenichel’s Invariant Manifold Theorem II: If > 0, but sufﬁciently small, there exists stable s u manifolds W (M) and unstable manifolds W (M) that lie within O() of, and are diffeo- s u morphic to, the stable manifold W (M0) and to the unstable manifold W (M0) respectively. Moreover, they are each locally invariant under (2.41), and Cr, including in , for any r < +∞.

Fenichel’s Invariant Manifold Theorem III: For every u ∈ M, there is an m- dimensional s s u stable manifold, W (u) ⊂ W (M), and an l-dimensional unstable manifold, W (u) ⊂ u s W (M), lying within O() of, and diffeomorphic to, the stable manifold W (u) and the unsta- u r ble manifold W (u) respectively. Moreover, they are C for any r, including in u and . The fam- s s s ily of stable manifolds {W (u): u ∈ M} is invariant in the sense that W (u)·D t ⊂ W (u· t), u if u· s ∈ D for all s ∈ [0, t], and the family of unstable manifolds {W (u): u ∈ M} is u u invariant in the sense that W (u)·D t ⊂ W (u· t), if u· s ∈ D for all s ∈ [t, 0].

Tikhonov Theorem: Consider the initial value problem

n x˙ = f(x, y, t) + ..., x(0) = x0, x ∈ D ⊂ < , t ≥ 0, n y˙ = g(x, y, t) + ..., y(0) = y0, y ∈ G ⊂ < , t ≥ 0.

For f and g, we take sufﬁciently smooth vector functions in x, y and t. 1. We assume that a unique solution of the initial value problem exists and suppose this holds also for the reduced problem

x˙ = f(x, y, t), x(0) = x0, 0 = g(x, y, t), with solution x˜(t), y˜(t). 2. Suppose that y˜ = Φ(x, t) is a solution of g(x, y, t) = 0, where Φ(x, t) is a continuous function and an isolated root. Also, suppose that y˜ = Φ(x, t) is an asymptotically stable solution of the dy + equation dτ = g(x, y, t) that is uniform in the parameters x ∈ D and t ∈ < . 3. y(0) is contained in an interior subset of the domain of attraction of y˜ = Φ(x, t) in the case of the parameter values x = x(0), t = 0. Then, we have

lim x(t) =x ˜(t), 0 ≤ t ≤ L, →0 lim y(t) =y ˜(t), 0 < d ≤ t ≤ L, →0 with d and L constants independent of .

35 2.3 Electric circuits

2.3.1 Linear and nonlinear electric elements

Linear electric elements are well known in many applications [9]. The main property of these elements is that their characteristics, that is, the relation between voltage υ and current i, are straight lines. One main linear electric element is the linear Resistor (Figure 2.9) for which the

Figure 2.9: Linear Resistor voltage (υ) across and the current satisfy the relation

υR = RiR (2.43) and its voltage- current (υ − i) characteristic is given in (Figure 2.10).

Figure 2.10: voltage- current (υ − i) characteristic of a linear resistor

An important linear electric element is the linear capacitor (Figure 2.11).The characteristic of the linear capacitor is given in (Figure 2.12). Another linear electric element is the linear inductor (Figure 2.13) where its characteristic is given in (Figure 2.14). The element for which the voltage (υ) across it and the current (i) satisfy

Figure 2.11: Linear capacitor

36 Figure 2.12: Characteristic of a linear capacitor

Figure 2.13: Linear inductor

Figure 2.14: Characteristic of a linear inductor the relation

RR = {(υ, i): f(υ, i) = 0}, (2.44) is called nonlinear electric element [5, 38]. There are many examples of nonlinear electric elements such as the nonlinear resistor, the nonlinear capacitor and the nonlinear inductor (Figure 2.15).

(a) nonlinear (b) nonlinear (c) nonlinear resistor capacitor inductor Figure 2.15: Nonlinear elements

37 Figure 2.16: Ideal diode

Figure 2.17: voltage- current (υ − i) characteristic of an ideal diode

A main example of nonlinear element is the ideal diode (Figure 2.16). The voltage- current (υ, i) characteristic of the diode consists of two straight lines (Figure 2.17) and is given by

RID = {(υ, i): υ· i = 0, i = 0 for υ < 0 and υ = 0 for i > 0}. (2.45)

If a diode is reverse biased (υ < 0), then the current is zero, i.e. the diode is an open circuit. If a diode conducts (i > 0), then the voltage is zero, i.e. the diode is a short-circuit. The power of an ideal diode is always zero. The elements with such a property are called non-energic elements. The ideal diode is an important circuit element for producing models of devices and circuits. Also plays an important role on the piecewise-linear analysis. In contrary to what happens in linear resistors the voltage - current (υ − i) characteristic is not a straight line passing through the origin. For this reason it is important that the symbol of a nonlinear resistor indicates its orientation. Also, the symbol of a non-linear resistor is not symmetrical to both its terminals. There are two kinds of independent sources in the theory of electric circuits and play the same role as the external forces on mechanics. The first kind of independent sources is an electric element for which the voltage in its terminals is a specific waveform υs(t) regardless of the current that flows through it i.e. the waveform is the same for any external circuit. This element is called the independent voltage source. The voltage, of the independent voltage source, is given by the relation

Rυs = (υ, i): υ = υs(t) for − ∞ < i < ∞. (2.46)

The voltage - current (υ − i) characteristic of this source is a straight line parallel to the current axis and can be considered as a nonlinear resistor controlled by the current. In the case when the straight line passes through the axes origin the voltage characteristic of the source

38 coincides with the characteristic of a short circuit. This property is very important in the analysis of electric circuits. The second kind of independent sources is an electrical element for which the current that passes through it has a speciﬁc waveform is(t) regardless the voltage on its terminals. This element is called the independent current source. The current, of the independent current source, is given by the relation

Ris = (υ, i): i = is(t) for − ∞ < υ < ∞. (2.47)

The voltage- current (υ − i) characteristic is a straight line parallel to the voltage axes and can be considered as a nonlinear resistor controlled by the voltage. In the case when the straight line passes through the axes origin, 0, the current source becomes an open circuit and its current is equal to zero.

2.3.2 Resistor connections and basic laws

As it is well known, resistors may connect in series or parallel. For the case where linear resistors are connected in series, the current that passes through each resistor is equal, the total voltage is the sum of the voltage accross each resistor and the total resistance is equal to the sum of the resistances of each individual series resistor. For nonlinear, controlled by current, resistors, the current that passes through the resistors is the same, i.e i = i1 = i2 = ... = in, where n the number of resistors and the voltage is equal to the sum of the voltages accross each individual nonlinear resistor, i.e. υ = υ1(i) + υ2(i) + ... + υn(i). For the case where linear resistors are connected in parallel the voltage across any given resistor is equal to the voltage across each of the other resistors, the total current is equal to the 1 sum of the currents that pass through the resistors and the total conductance, G = R , is equal to the sum of the conductance of each resistor. For nonlinear, controlled by voltage, parallel resistors, the voltage across any given resistor is equal to the voltage across each of the other resistors and the total current is equal to the sum of the currents that pass through the resistors, i.e. i(υ) = i1(υ) + i2(υ) + ... + in(υ), where n is the number of the resistors.

A branch represents a single element such as resistor, source etc. The point of connection between two or more branches is called a node. Any closed path in a circuit is called a loop.

Kirchhoff’s ﬁrst law: The algebraic sum of currents entering and leaving a node is zero, P in = 0.

P Kirchhoff’s second law: The algebraic sum of all voltages around a loop is zero, υn = 0

39 Figure 2.18: Differential Ampliﬁer

Figure 2.19: Operational Ampliﬁer

2.3.3 Operational Ampliﬁer

The operational amplifier is a basic element for the design of an analog electric circuit and plays important role for executing certain operations such as division, addition or integration. There is a great variety of operational amplifiers that one can use in order to implement different kinds of systems. A differential amplifier (Figure 2.18) enhances the differential input. Indeed the output voltage is equal to the enhanced of the differential input voltage and is given by the relation

υ0 = Avid, (2.48)

where υid = ν+ − ν−. A differential amplifier with infinite voltage gain is called Operational Amplifier (Figure 2.19). The ideal operational amplifier is a special case of the ideal differential amplifier where the differential input resistance is infinite, RID = ∞, the output resistance is zero, Ro = 0, and most important the gain is infinite, A = ∞. The function that an operational amplifier will carry out is determined by the way it will be connected with the other electric elements of the circuit. Therefore an operational amplifier may act as a reversing or non reversing amplifier, as a buffer, as an adder, as a sub-tractor, as a filter, or as an integrator/ differentiator.

40 2.3.4 Examples of nonlinear circuits

As we will show, nonlinear electric circuits may implement nonlinear systems of differential equations. Such circuits have rich dynamical behavior and cover all the known routes to chaos [34, 40, 41, 52, 62, 67].

A second order electric circuit with nonlinear capacitor

The nonlinear capacitor is a varactor diode with characteristic

p Q = −k V0 − υc = Q(υc). (2.49)

For values υc < V0 the diode behaves as a nonlinear capacitor while for υc > V0 behaves as a nonlinear resistor [40, 67]. The circuit (Figure 2.20) implements the second order differential equation

d2q dq + r + (eq − 1) = B cos ω τ, (2.50) dτ 2 dτ N

1 1 1 Q 1 C0 E0 where q = V0, τ = t( ) 2 , r = R( ) 2 ,B = , ωN = ω(C0L) 2 . C0 C0L L V0

Figure 2.20: A second order electric circuit with nonlinear capacitor

From the bifurcation diagrams (Figure 2.21) we conclude that the above circuit may perform periodic or chaotic oscillations (Figure 2.22).

(a) r = 0.4, f = 0.16,B = 1.5 (b) r = 0.4, f = 0.16,B = 1.5 Figure 2.21: Bifurcation diagrams for the second order electric circuit with nonlinear capacitor

41 (a) r = 0.4, f = 0.16,B = 1.5 (b) r = 0.4, f = 0.16,B = 1.5

(e) r = 0.4, f = 0.16,B = 6 (f) r = 0.4, f = 0.16,B = 6 Figure 2.22: Trajectories and phase portraits for the second order electric circuit with nonlinear capacitor

A second order electric circuit with nonlinear inductor

An example of a second order electric circuit with nonlinear inductor is given in Figure 2.23. This circuit implements a Dufﬁng type differential equation

d2x dx + + ax + bx3 = B cos ωt, (2.51) dt2 dt

1 E0 where x = φ is the magnetic ﬂux, = RC , a, b and B = RC are parameters.

42 Figure 2.23: A second order electric circuit with nonlinear inductor [38]

As we see from the bifurcation diagram (Figure 2.24) the system is driven to chaos through period doubling. As the parameter B increases the system goes to period-1 state following a reverse sequence of period doubling.

(a) ω = 0.8, = 0.18, a = 1.0, b = 1.0 Figure 2.24: Bifurcation diagram for the second order electric circuit with nonlinear inductor

A second order electric circuit with nonlinear resistor

An example of a second order electric circuit with nonlinear resistor is given in Figure 2.25. This circuit implements another Dufﬁng type differential equation d2x dx − µ(1 − x2) + x3 = B cos ω τ, (2.52) dτ 2 dτ N

q q 2 υc 3L 3(L−R C) called Dufﬁng-van der Pol [38, 67], where x = , a = 2 , µ = 2 ,B = V0 L−R C R C 3 √ a E0 t , ωN = aω LC and τ = √ . V0 a LC The voltage - current (υ, i) characteristic of the nonlinear resistor is a smooth cubic characteristic given by the relation 2 1 υc i = − υc(1 − 2 ), (2.53) R Vs

43 Figure 2.25: A second order electric circuit with nonlinear resistor

where Vs is a constant parameter. From the bifurcation diagram (Figure 2.26) we see that the system may oscillate in a regular way, periodically or semi periodically (Figure 2.27-a) or make chaotic oscillations (Figure 2.27-b).

(a) ω = 0.7, µ = 0.2 Figure 2.26: Bifurcation diagram for the second order electric circuit with nonlinear resistor

(a) ω = 0.7, µ = 0.2,B = 1.2 (b) ω = 0.7, µ = 0.2,B = 1.38 Figure 2.27: Poincare sections for the Dufﬁng- van der Pol electric circuit with nonlinear Resistor

44 Non- autonomous electric circuit with nonlinear resistor of type- N

The circuit is given in Figure 2.28[41] and implements the system of differential equations dυ 1 C = (i − i ), dt C L N di 1 L = (−Ri − υ + V sin ωt), (2.54) dt L L C 0 where iN = GbvC + 0.5(Ga − Gb)(|VC + Bp| − |VC − Bp|) is the characteristic of the nonlinear resistor (Figure 2.29).

Figure 2.28: Electric circuit with nonlinear resistor of type-N

Figure 2.29: Characteristic of the nonlinear resistor

The above circuit is implemented experimentally for the values L = 32.9mH, R = 700Ω,C =

62.9nF, Ga = −2.2mS, Gb = 1.0mS and Bp = 1.878V in Figure 2.30. As we can see in the bifurcation diagrams (Figure 2.31) the system has rich dynamics. The 2π bifurcation diagram changes drastically with the variation of the frequency f = ω . The system is lead to chaos in a very different way, i.e. there are periodic areas, where the period is increased sequentially, and chaotic areas between them. This way to chaos is called “period-adding route to chaos”. Also, the bifurcation diagram starts with a sudden transition from period one to chaos and ends with a sudden transition from chaos to period one.

45 Figure 2.30: Experimental implementation of a type N nonlinear resistor

(a) f = 2500Hz (b) f = 5000Hz Figure 2.31: Bifurcation diagrams of the electric circuit with a nonlinear resistor of type-N

46 Chapter 3

Classiﬁcation of the Slow Invariant Manifold

In mechanical applications, there is a great interest in structures of linear systems with nonlinear attachments [15, 12, 55, 68, 63]. In most of these studies the nonlinear attachments have small masses in comparison to the structures to which they are attached, and the systems are non conservative. Various dynamical phenomena have been investigated for two degrees of freedom damped systems with or without external forcing. A special property of these configurations is that the nonlinear substructures can act as nonlinear energy sinks (NESs) and absorb, through irreversible transient transfer, energy from the linear parts. There has been shown, both numerically and analytically, that the above can occur through resonance captures in vicinities of resonance manifolds of the underlying conservative systems [12, 63] for certain ranges of parameters and initial conditions. In the case of external forcing various other complicated dynamical phenomena may appear [15, 16, 63]. These systems can also be studied with the help of the method of multiple scales and singularity analysis [15, 16, 18, 19, 63, 69]. It has been shown that the Slow Invariant Manifolds (SIMs) obtained in the singular limit plays a very important role in the dynamics under consideration. If a SIM is stable it can attract the orbits of the system in the limit of ’fast’ time scales for a wide range of initial conditions and accounts for transient energy transfer. If it is unstable, for example of saddle type, it has locally invariant manifolds and their existence affect the dynamics of such systems and may produce relaxation oscillations and complex phenomena [18, 19, 33, 50]. In what follows we study a three degree of freedom autonomous system considered previously in [55, 68]. Through singular transformations and reduction of the dynamics, we derive a reduced non autonomous damped strongly nonlinear second order differential equation. With the use of the complexification-averaging technique (CX-A) we obtain a system of two, first order, differential equations governed by the slow time scale. The SIM of the above system, computed through multiple scale analysis or singularity analysis [69], provides information about the asymptotic behavior of the orbits of the reduced system. The SIM of the system may have

47 one or three branches depending on the slow time and the value of the damping parameter. The corresponding bifurcations affect the dynamical behavior of the system which can change drastically. Indeed, Tikhonov’s theorem guarantees that, when the branches of the SIM are isolated and stable, the orbits of the system tend to these stable branches as long as they exist. The application of Tikhonov’s theorem allows us to classify the behavior of the orbits in comparison to the evolution of the SIM, depending on the system parameters. The theoretical analysis provides the range of the system parameters for each of the previous cases and allows us to predict the long term behavior of the orbits of the reduced system. In fact, the damping parameter λ and the number of the branches of the SIM play an essential role in the evolution of the dynamics of the system. The dynamics can be simple, as in the case where there is only one persisting stable branch of the SIM, or complicated, when bifurcations of the SIM occur, resulting in different phenomena such as relaxation oscillations or orbit excitations. The chapter is organized as follows. In the first section, we present the system and its reduction through singular transformations and the complexification averaging technique(CX-A). In the second section, we apply multiple scale analysis to the resulting system and compute the SIM. The polynomial, that defines it, depends on the slow time. We study the change of the number of its roots, depending on time and system parameters, and determine the bifurcations and stability of the different branches of the SIM. Based on Tichonov’s theorem we find the range of parameters and the interval of time where the branches of the SIM are attracting. In the fifth section, we present numerical results and classify the behavior of the orbits. Finally we present some concluding remarks.

3.1 Reduction of the system

The initial system considered in this work is composed of two coupled linear oscillators and a nonlinear oscillator that interacts through an essential nonlinearity with one of the linear oscillators. The mass of the nonlinear attachment is small in comparison to the masses of the linear oscillators and therefore the problem is singular. The governing equations of motion are given by,

3 y¨ + λ(y ˙ − x˙ 0) + C(y − x0) = 0, 3 x¨0 + d(x0 − x1) = λ(y ˙ − x˙ 0) + C(y − x0) ,

x¨1 + ax1 + d(x1 − x0) = 0, (3.1)

where y, x0, x1 are the displacements and a, d, C, λ and << 1 are the parameters of the system. After applying the linear singular transformation

−1/2 1/2 v = x0 + y, −1/2 −1/2 w = x0 − y, (3.2)

48 the system assumes the form: v + w w¨ + λ(1 + )w ˙ + C(1 + )w3 = −d + dx , 1 + 1 d dw v¨ + v − dx = − , 1 + 1 1 + d dw x¨ + (a + d)x − v = . (3.3) 1 1 1 + 1 + The second and third equations of (3.3) form a non homogeneous set of linear ordinary differential equations with constant coefﬁcients, with eigenfrequencies

2 ω1 = (1 + )a + (2 + )d − p−4ad(1 + ) + ((1 + )a + (2 + )d)2 , 2(1 + ) 2 ω2 = (1 + )a + (2 + )d + p−4ad(1 + ) + ((1 + )a + (2 + )d)2 . 2(1 + ) (3.4) and corresponding eigenvectors (K1, 1) and (K2, 1), where

K1 = (1 + )a + d − p−4ad(1 + ) + ((1 + )a + (2 + )d)2 , 2d K2 = (1 + )a + d + p−4ad(1 + ) + ((1 + )a + (2 + )d)2 . (3.5) 2d We introduce modal coordinates for the linear part of (2) through the coordinate transformations, v = K1z1 + K2z2, x1 = z1 + z2, assume zero initial displacements and nonzero initial velocities z˙1(0) = ω1z10, z˙2(0) = ω2z20, and obtain the approximate system

z1(t) = z10sinω1t + O(),

z2(t) = z20sinω2t + O(), dK w¨ + λ(1 + )w ˙ + C(1 + )w3 = (d − 1 )z sinω t 1 + 10 1 dK + (d − 2 )z sinω t + O(). (3.6) 1 + 20 2 Combining the above equations we obtain the following approximate reduced system

3 w¨ + λ(1 + )w ˙ + C(1 + )w = Asinω1t + Bsinω2t + O(), (3.7) where we have replaced the solutions of the non homogeneous linear oscillators and where dK A = (d − 1 )z , 1 + 10 dK B = (d − 2 )z . (3.8) 1 + 20

49 ¯ After performing a time transformation tˆ → ω20t, where ω2 = ω20 + B, the above equation becomes ω B¯ w00 + λwˆ 0 + Cwˆ 3 = Asinˆ ( 1 tˆ) + Bsinˆ (1 + tˆ) + O(), (3.9) ω20 ω20 where ˆ C(1 + ) ˆ λ(1 + ) ˆ A ˆ B C = 2 , λ = , A = 2 , B = 2 . (3.10) ω20 ω20 ω20 ω20 Considering the reduced system (3.9) we apply the complexification-averaging technique [46, 63]. We introduce the complex coefficient Ψ = w0 + jw, so j 1 j w = − (Ψ − Ψ∗), w0 = (Ψ + Ψ∗), w00 = Ψ0 − (Ψ + Ψ∗), (3.11) 2 2 2 where Ψ∗ is the conjugate. Then equation (3.9) yields λˆ j Cjˆ ω B¯ Ψ0 + ( − )(Ψ + Ψ∗) + (Ψ − Ψ∗)3 = Asinˆ ( 1 tˆ) + Bsinˆ ((1 + )tˆ). (3.12) 2 2 8 ω20 ω20 At this point we make the important additional assumption that the considered dynamics is dominated by a single normalized ’fast’ frequency equal to unity. It follows that our results will only be valid as long as this assumption is satisfied. Mathematically, this assumption is imposed by expresssing Ψ = ϕejt so that equation (3.12) becomes λˆ j φ0 + jφ + ( − )(φ + φ∗e−2jtˆ) 2 2 Cjˆ + (φ3e2jtˆ − 3φ2φ∗ + 3φ(φ∗)2e−2jtˆ − (φ∗)3e−4jtˆ) 8 ˆ ω −ω −ω −ω ˆ ¯ ¯ jA j 1 20 tˆ j 1 20 tˆ Bj j B tˆ −j(2+ B )tˆ = − (e ω20 − e ω20 ) − (e ω20 − e ω20 ). (3.13) 2 2 We mention at this point that system (3.13) is approximate since it takes into account only a single ’fast’ frequency. Clearly, since the system under consideration (and the reduced system (3.9)) is strongly nonlinear, this is only an approximation since higher harmonics will exist in the dynamics. However, we conjecture that there exist regimes where the harmonic components with normalized frequency unity dominate [46, 63]. If we average out the harmonic terms with fast frequencies equal to unity or multiples of it, by integrating (3.13) over the common period 2π of the integral periods, and leaving out O() terms of the above equation, we derive the following averaged complex dynamical system

ˆ ˆ ˜ ˆ ¯ 0 λ j 3Cj 2 J jA Bj j B tˆ φ + ( + )φ − |φ| φ + + + e ω20 = 0, (3.14) 2 2 8 2 2 2 where ˆ Aω20ω1 ω1 J = 2 2 (cos( 2π) − 1), π(ω1 − ω20) ω20 ˆ 2 ˜ Aω20 ω1 A = 2 2 sin( 2π). (3.15) π(ω1 − ω20) ω20

50 Since we are interested in the amplitude of the oscillations of the nonlinear attachment, we use the polar representation of the complex number ϕ = N(t)ejn(t). After separating the real and imaginary parts, we obtain the following two equations for N and η, that represents the slow ﬂow of the system

λNˆ J A˜ Bˆ B¯ N 0 = − − cos(η) − sin(η) + sin( tˆ− η), 2 2 2 2 ω20 N 3CNˆ 3 J A˜ Bˆ B¯ Nη0 = − + + sin(η) − cos(η) − cos( tˆ− η). (3.16) 2 8 2 2 2 ω20 This reduced system will be the basis for the asymptotic analysis that will be carried out in the next sections.

3.2 Multiple scales and singular perturbation analysis

We use the multiple scales analysis [15, 16, 46, 55, 68, 63]

N(t) = N(t0, t1, ....)

= N0(t0, t1, ...) + N1(t0, t1, ...) + O(),

η(t) = η(t0, t1, ....)

= η0(t0, t1, ...) + η1(t0, t1, ...) + O(), (3.17) where t0 = tˆand t1 = tˆ. By keeping O(1) terms in(3.16) for N and η we derive the equations

ˆ ˜ ˆ ∂N0 λN0 J A B ¯ + + cos(η0) + sin(η0) − sin(Btˆ− η0) = 0, ∂t0 2 2 2 2 ˆ 3 ˜ ˆ ∂η0 N0 3CN0 J A B ¯ N0 + − − sin(η0) + cos(η0) + cos(Btˆ− η0) = 0. (3.18) ∂t0 2 8 2 2 2

To study the steady state dynamics of the above system, in terms of the fast time scale t0, we

∂Nˆ0 ∂n0 examine the limit of the dynamics as t0 → ∞ and impose the conditions = 0, = 0. This ∂t0 ∂t0 will provide us with the long-term behavior of the dynamics in the limit of large values of the fast time scale. Then, from equations (3.18) we ﬁnd

λˆNˆ J A˜ Bˆ 0 = − cos(ˆη ) − sin(ˆη ) + sin(B¯tˆ− ηˆ ) ≡ Σ¯ , 2 2 0 2 0 2 0 1 Nˆ 3CˆNˆ 3 J A˜ Bˆ 0 − 0 = sin(ˆη ) − cos(ˆη ) − cos(B¯tˆ− ηˆ ) ≡ Σ¯ . (3.19) 2 8 2 0 2 0 2 0 2 Manipulating expressions (3.19) we derive the steady state phase

¯ 3CNˆ 3 ¯ ˜ ˆ B 0 ˆ ˆ ˆ ˆ B (A + B cos( tˆ))( − N0) + λN0(B sin( tˆ) − J) ω20 4 ω20 cosη ˆ0 = , J 2 + A˜2 + 2A˜Bcosˆ ( B¯ tˆ) + Bˆ2 − 2JBˆ sin( B¯ tˆ) ω20 ω20 (3.20)

51 with the steady state amplitude given by

8 16(λˆ2 + 1) Nˆ 6 − Nˆ 4 + Nˆ 2 = 0 3Cˆ 0 9Cˆ2 0 16 B¯ (A˜2 + 2AˆBˆ cos( tˆ) + Bˆ2 9Cˆ2 ω20 B¯ +J 2 − 2JBˆ sin( tˆ)). (3.21) ω20

Equations (3.20) and (3.21) represent the slow invariant manifold (SIM) of the dynamics of (3.9) [15, 63] in terms of the fast time scale (note that the slow time scale still appears in coefﬁcients of equation (3.21)). In order to study the bifurcations of the SIM we reconsider equations (3.19). After taking 3Cˆ ˆ 2 the squares of both equations, adding them and introducing the new variable z = 1 − 4 N0 we derive the equation

3Cˆ P (z) = z3 − z2 + λˆ2z − λˆ2 + Σ¯ = 0, (3.22) 4 where Σ¯ = A˜2 + 2A˜Bˆ cos( B¯ tˆ) + Bˆ2 + J 2 − 2JBˆ sin( B¯ tˆ). From the derivative of the above ω20 ω20 cubic polynomial P 0(z) = 3z2 − 2z + λˆ2, (3.23) we see that, for λˆ > √1 , P (z) is a monotone increasing function and therefore has only one real 3 √ 1± 1−3λˆ2 root, while for λˆ < √1 , P 0(z) equals zero for z = , and depending on the value of Σ¯ 3 3 the polynomial P (z) may have one or three real roots (Figure 3.1). When 2 3 2 81C (1 − 3λˆ ) 2 > |1 + 9λˆ − Σ¯|, (3.24) 8 it has three real roots, otherwise only one. The roots Nˆ 2 = 4 (1 − z ) are positive. We note that 0 3Cˆ 0 when 2 81C 2 3 |1 + 9λˆ − Σ¯| = (1 − 3λˆ ) 2 , (3.25) 8 for a certain value of time, two of the three real roots are equal and saddle-node bifurcations occur. The above analysis with multiple scales is equivalent to singularity analysis by taking the slow time tˆ→ B¯tˆ. Then system (3.16) becomes

N 0 = f(N, n, tˆ), Nn0 = g(N, n, tˆ) (3.26) and at the singular limit ( = 0) we have f = g = 0 which is exactly the SIM of the system. Tichonov’s theorem [59, 69] guarantees that, when the roots of f = g = 0 are isolated and at the

52 same time there exist stable solutions of

dN = f(N, n, tˆ), dτ dn N = g(N, n, tˆ), (3.27) dτ

ˆ with tˆconsidered as a parameter, then N(tˆ) → N0(tˆ), η(tˆ) → ηˆ0(tˆ) as → 0. In order to ﬁnd when the SIM is stable, according to Tichonov’s theorem, we ﬁnd the linear stability of the roots of (3.19) when they are isolated. In the case when (3.25) holds the two real roots are not isolated and the time when the bifurcation occurs does not fall under Tichonov’s theorem. When the solutions are away from the bifurcation point the matrix of the linearized system around the solution of the SIM is

λˆ J A˜ Bˆ ¯ˆ ! − 2 2 sin(ˆη0) − 2 cos(ˆη0) − 2 cos(Bt − ηˆ0) 1 9Cˆ ˆ 2 J A˜ Bˆ ¯ˆ − 2 + 8 N0 2 cos(ˆη0) + 2 sin(ˆη0) − 2 sin(Bt − ηˆ0)

ˆ where N0 and ηˆ0 are the roots of (3.19). ¯ ¯ ∂Σ1 ¯ ∂Σ2 ¯ Since = Σ2, = −Σ1 the characteristic polynomial of the linearized system becomes ∂ηˆ0 ∂ηˆ0

λˆ λˆ 1 9Cˆ µ2 + (Σ¯ + )µ + Σ¯ + ( − N 2)Σ¯ = 0. (3.28) 1 2 2 1 2 8 0 2

Where µ are the eihenvalues of the above matrix. When λˆ Σ¯ + > 0, (3.29) 1 2 and 9Cˆ λˆΣ¯ + (1 − Nˆ 2)Σ¯ > 0, (3.30) 1 4 0 2 the two roots µ1, µ2 of the characteristic polynomial have negative real parts. Using (3.19) and the variable z,(3.30) becomes 3z2 − 2z + λˆ2 > 0 (3.31) which is the derivative of P (z), calculated at the roots of the deﬁning polynomial of the SIM (3.22). For λ > √1 ,P 0(z) is greater than zero for all z and therefore the SIM is stable. For 3 λˆ < √1 ,P 0(z) is greater than zero outside its roots which are at the same time the maximum 3 and minimum of the polynomial (3.22). Therefore the solution of P (z), that lies between the minimum and the maximum is unstable, while the lower and the higher roots (when they exist) 8 3 ¯ ˆ2 ˆ2 2 are stable. Therefore for Σ < 81C ((1 + 9λ ) − (1 − 3λ ) ) only the lower solution exists and is stable. The solutions N(tˆ), η(tˆ) of (3.16) tend to it. As time increases and the above inequality is not satisﬁed, two more roots appear through a saddle-node bifurcation and again disappear, 8 3 ¯ ˆ2 ˆ2 2 when Σ > 81C ((1 + 9λ ) + (1 − 3λ ) , and for this time interval only the higher root exists.

53 q ˆ 1 (a) λ < 3

q ˆ 1 (b) λ > 3 Figure 3.1: The polynomial P (z) for different values of λˆ

3.3 Behavior of the solutions and numerical results

Since we have found that, the steady state solutions that are lower or higher than the minimum and maximum of the polynomial P (z), are linearly asymptotically stable, i.e. hyperbolic points, they preserve their stability in the nonlinear system and have a basin of attraction for the interval of time that they exist. According to Tichonov’s theorem the orbits of the slow ﬂow (3.16) tend to these stable branches of the SIM for the above interval of time. Therefore we perform numerical integration of (3.16) for different values of the parameters A,˜ B,ˆ , λˆ and Cˆ = 2 in order to study what is the relative position of the amplitude of the orbit of the slow ﬂow after some time in comparison with the position of the SIM. First we consider the case when the SIM has a single branch. The conditions for this case are: (i) λˆ > √1 , 3 1 8 2 2 3 or (iia) λˆ < √ and Σ¯ < ((1 + 9λˆ ) − (1 − 3λˆ ) 2 ), for all time, 3 81Cˆ 1 8 2 2 3 or (iib) λˆ < √ and Σ¯ > ((1 + 9λˆ ) + (1 − 3λˆ ) 2 ), for all time. 3 81Cˆ From the above theoretical analysis, we expect that, for these conditions since we have no

54 bifurcations, the orbits of the slow ﬂow tend to the stable SIM. This is conﬁrmed by direct numerical simulation of system (3.16) which is depicted in Figure 3.2 and compared to the asymptotic solution predicted by the SIM (3.21) or (3.22).

(a) A˜ = 0.0748516, Bˆ = 0.674394,J = 0.00666891, Cˆ = 2, = 0.001, λˆ = 0.8

(b) A˜ = 1, Bˆ = 1,J = 1, Cˆ = 2, = 0.001, λˆ = 0.2 Figure 3.2: The case where the SIM has always one stable branch (gray solid line: SIM, black dashed line: N(t) of (3.16))

We now consider the second case when the SIM has three branches, two of which are stable and one is unstable. The condition for this case is: 1 8 2 2 3 8 2 2 3 λˆ < √ and ((1 + 9λˆ ) − (1 − 3λˆ ) 2 ) < Σ¯ < ((1 + 9λˆ ) + (1 − 3λˆ ) 2 ). 3 81Cˆ 81Cˆ In this case we also have no bifurcations, and for all time the SIM possesses three branches (Figure 3.3). For different initial conditions the orbits of the slow ﬂow either tend to the upper or to the lower stable branches of the SIM. The third case is when relaxation oscillations occur, with the dynamics making transitions between the two stable branches of the SIM. In this case bifurcations occur as the orbits of the dynamical system undergo transitions between all branches of the SIM. For a certain period of the slow time scale only the upper branch exists. Then, after a saddle-node bifurcation, two additional branches appear, the one stable and the other unstable. This phenomenon takes place

55 (a) A˜ = 0.0129, Bˆ = 0.2956,J = 0.033, Cˆ = 2, = 0.01, λˆ = 0.3

(b) A˜ = 0.01, Bˆ = 0.2,J = 0.01, Cˆ = 2, = 0.001, λˆ = 0.1 Figure 3.3: The case where the SIM has always three branches (gray solid line: SIM, black dashed line: N(t) of (3.16))

for a period of the slow time scale that depends on the parameters A,˜ B,ˆ λˆ and C.ˆ Then the stable upper branch and the unstable branch of the SIM coalesce through saddle-node bifurcations. Therefore, the orbits of (3.16) that are initially attracted to the upper branch of the SIM, make a sudden transition (jump) to the lower branch producing relaxation oscillations (Figure 3.4). After sufﬁcient increase of the slow time scale the upper branch reappears and the lower one disappears again through a saddle node bifurcation. This behavior is repeated periodically and gives rise to sustained relaxation oscillations in the dynamics. These relaxation oscillations indicate strong transient energy transfer in our reduced system, from the system of linear coupled oscillators to the essentially nonlinear attachment. In the fourth case bifurcations can occur either, between the upper stable branch of the SIM and the unstable branch, or only between the lower stable branch of the SIM and the unstable branch, resulting into the disappearance of a pair of stable-unstable branches of the SIM for a certain period of the slow time scale. From the analysis of section 3 we expect that the corresponding orbit will be captured in the persisting stable branch of the SIM and this is

56 (a) A˜ = 0.091, Bˆ = 0.3463,J = 0.0189, Cˆ = 2, = 0.01, λˆ = 0.4

(b) A˜ = 0.25, Bˆ = 0.15,J = 0.1, Cˆ = 2, = 0.2, λˆ = 0.4 Figure 3.4: Relaxation oscillations (gray solid line: SIM, black dashed line: N(t) of (3.16)) confirmed by the numerical simulations depicted in Figure 3.6. We note here that, when the structur of the SIM changes from the case of relacation oscillations to the present case, for these values of the damping parameter λˆ, the slow flow may still perform relaxation oscillations (Figure 3.5) The fifth case corresponds to the excitation of periodic or non periodic orbits of the slow flow for certain initial conditions. For sufficiently small values of damping λˆ the orbits of (3.16) may oscillate rapidly, while approaching the stable branch of the SIM, and follow it until the time where there is bifurcations between the stable and the unstable brances of the SIM. Then an orbit is excited and the entire phenomenon it repeated (Figures 3.7).

3.4 Conclusions

The study of a three degree of freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment was performed by computing the Slow Invariant Manifold (SIM) of a reduced non-autonomous second order differential equation and studying the stability

57 steady state solutions 1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 t 0 500 1000 1500 2000 Figure 3.5: Relaxation Oscillations of the slow ﬂow when the SIM has the structure of case 4, A˜ = −0.093, Bˆ = −0.175,J = 0.105, Cˆ = 2, = 0.1, λˆ = 0.09

(a) A˜ = 0.139, Bˆ = 0.283,J = 0.0369, Cˆ = 2, = 0.01, λˆ = 0.3

(b) A˜ = 0.1, Bˆ = 0.1,J = 0.1, Cˆ = 2, = 0.01, λˆ = 0.1 Figure 3.6: Bifurcations between the upper (or the lower) stable branch and the unstable branch of the SIM (gray solid line: SIM, black dashed line: N(t) of (3.16))

58 ˆ (a) A˜ = 1, Bˆ = 1,J = 0.1, Cˆ = 2, = 0.01, λ = 0.023, ω20 = 1, B¯ = 0.01

(b) A˜ = 0.165319, Bˆ = 0.149,J = 0.009289, Cˆ = 2, = ˆ 0.01, λ = 0.023, ω20 = 1, B¯ = 0.01

(c) A˜ = −0.127, Bˆ = 0.139,J = 0.125, Cˆ = 2, = 0.1, λˆ = 0.059, ω20 = 1, B¯ = 0.0328 Figure 3.7: Excitation of the orbits (gray solid line: SIM, black dashed line: N(t) of (3.16)) and bifurcations of this manifold as the system parameters vary. Depending on the parameters of the system, we have shown through dynamical analysis and Tikhonov’s theorem, that the SIM can either have one branch that is stable, or three branches, two which are stable and one unstable. For relatively large values of the damping parameter, the structure of the SIM is simple and the orbits of the system are attracted by it.

59 The interplay between the stable and the unstable branches of the SIM produces interesting dynamical phenomena of the slow flow such as orbit captures, relaxation oscillations, excitations of periodic orbits and possible chaotic orbits. Orbit capture occurs when there is one or two persisting stable branches of the SIM (i.e., stable SIM branches that are not eliminated through bifurcations as slow time increases). In the case when a single persisting stable branch of the SIM exists, it seems to be globally attractive and the long term behavior of the orbits is simple, since they simply tend to it with increasing time. In the case where one of the stable branches persists and the other two branches bifurcate (e.g., Figure 3.6) the orbit always relaxes to the persistent stable branch. This behavior can be predicted in terms of the parameters defining the SIM A,˜ B,ˆ λˆ and Cˆ in equations (3.21) and (3.22). Relaxation oscillations occur when we have bifurcations of all three branches of the SIM and this type of dynamics is related to energy transfer of the system, from the linear oscillators to the essentially nonlinear attachment. In the fifth case, the excitation of orbit may indicate a possible chaotic behavior of the slow flow of the system. This possibility is investigated in the next chapter. The exact conditions required for the SIM to have various topologies were found in this work analytically. This allows us to predict the values of the system parameters for which the dynamics have a certain type of behavior.

60 Chapter 4

The dynamics of the slow ﬂow

In the previous chapter we showed that the structure of the SIM is related to the parameters and the initial conditions of the system, specially the damping parameter λˆ. We studied the different cases of the SIM and its bifurcations as time evolves. We also showed that several phenomena appears in the slow flow such as relaxation oscillations, orbit excitations and possible chaotic behavior, especially when the damping parameter is small. On this chapter we study the dynamical behavior of slow flow of the system, investigating the possible chaotic behavior and the different relaxation oscillations. From the numerical simulations, for a wide variety of parameters, we show that the slow flow has rich dynamics, depending in the parameters of the system, with relaxation oscillations, regular orbits and chaotic behavior.

4.1 The dynamics of the slow ﬂow

We consider the reduced system (3.9)

ω B¯ w00 + λwˆ 0 + Cwˆ 3 = Asinˆ ( 1 tˆ) + Bsinˆ (1 + tˆ) + O(), ω20 ω20 where

ˆ C ˆ λ ˆ A ˆ B C = 2 , λ = , A = 2 , B = 2 , ω20 ω20 ω20 ω20 A = (d − dK1)z10,B = (d − dK2)z20.

After applying the complexiﬁcation- averaging technique [46, 63] by introducing the complex coefﬁcient Ψ = w0 + jw and make the important additional assumption that the considered dynamics is dominated by a single normalized ’fast’ frequency equal to unity, i.e. Ψ = ϕejt, we derive the averaged complex dynamical system

ˆ ˆ ˜ ˆ ¯ 0 λ j 3Cj 2 J jA Bj j B tˆ φ + ( + )φ − |φ| φ + + + e ω20 = 0, 2 2 8 2 2 2

61 where ˆ Aω20ω1 ω1 J = 2 2 (cos( 2π) − 1), π(ω1 − ω20) ω20 ˆ 2 ˜ Aω20 ω1 A = 2 2 sin( 2π). π(ω1 − ω20) ω20 With the use of the polar representation of the complex number ϕ = N(t)eiη(t), after separating the real and imaginary parts, we obtain the following two equations for the amplitude N and the phase η. λNˆ J A˜ Bˆ B¯ N 0 = − − cos(η) − sin(η) + sin( tˆ− η), 2 2 2 2 ω20 N 3CNˆ 3 J A˜ Bˆ B¯ Nη0 = − + + sin(η) − cos(η) − cos( tˆ− η). 2 8 2 2 2 ω20 This represents the slow ﬂow for the N and η variables. For convenience in the numerical simulations we do not use the amplitude N and the phase

η of φ but the real x and the imaginary px parts of φ = x + jpx λ p 3C J Bˆ B¯ 0 x 2 2 ˆ x = − x + − (x + px)px − + sin( t) 2 2 8 2 2 ω20 λ x 3C A˜ Bˆ B¯ 0 2 2 ˆ px = − px − + (x + px)x − − cos( t). (4.1) 2 2 8 2 2 ω20 In the previous chapter we found analytically that the SIM may have one branch that is always stable, three branches, where the two are stable and the one that lies between them unstable, or may have bifurcations between all of them. In order to study the dynamics of the slow flow we take numerical simulations for different initial conditions and values of the parameters. We calculate the bifurcation diagram for certain initial condition and different values of the damping parameter λˆ. The SIM and the the slow flow is given for certain values of λˆ. In order to study if there is chaotic behavior of the slow flow we calculated the maximal Lyapunov Characteristic Exponent [56] and the Poincare section for many different initial conditions. We made a high number of simulations, presenting here some characteristic examples. The overall behavior of the slow flow is the following: Regular Orbits. In this case, the slow flow oscillates regularly in the region of the stable branch of the SIM. As predicted by Tikhonov’s theorem if there is one stable or two separate stable branches of the SIM the slow flow will fall in the basins of attraction of one of the stable branches of the SIM. For example, for the parameters A˜ = −0.075, Bˆ = 0.614,J = 0.063, = 0.1, B¯ = ω20 −0.138, for different values of the damping parameter λˆ, the slow flow has only regular orbits. This is what we expected because the SIM has only one stable branch and the slow flow oscillates regularly in the region near this branch with always negative values of the maximal Lyapunov Characteristic Exponent (Figure 4.1).

62 The Poincare section shows that, in the limiting process, the orbits are attracted to a periodic one. The same behavior is seen for several values of the parameters, where the SIM has only one stable branch (Figure 4.2). The same behavior is seen for the parameters A˜ = −0.113, Bˆ = 0.124,J = 0.135, = 0.1, B¯ = −0.073, λˆ = 0.01, (Figure 4.3-a,b) where the SIM has three branches and the slow ω20 flow oscillates in one of the stable branches of the SIM. The negative maximal Lyapunov Exponent and the Poincare sections, show that the slow flow has only regular orbits in the above case. When λˆ increase, for example for λˆ = 0.2, the structure of the SIM change and the system undergoes saddle - node bifurcations (Figures 4.3- c,d). As can be seen, although the system undergo saddle - node bifurcation, the orbits of the slow flow oscillates in the region of the stable branch of the SIM and no relaxation oscillation occurs in contrast to what we will see in the next case. In both cases the orbits of the slow flow are regular.

63 (a) Bifurcation diagram

steady state solutions 1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 t 0 200 400 600 800 1000 (b) SIM and slow ﬂow for λˆ = 0.05

mLCE

0.04

0.02

0.00 t 5000 10 000 15 000 20 000

-0.02

(d) maximal LCE for λˆ = 0.05 Figure 4.1: A˜ = −0.075, Bˆ = 0.614,J = 0.063, = 0.1, B¯ = −0.138 ω20

64 steady state solutions mLCE 0.010 1.4 0.005 1.2

1.0 0.000 t 10 000 20 000 30 000 40 000 50 000 0.8 -0.005 0.6 -0.010 0.4

0.2 -0.015

0.0 t 0 500 1000 1500 2000-0.020 (a) SIM and slow ﬂow for A˜ = 0.0168, Bˆ =(b) maximal LCE for A˜ = 0.0168, Bˆ = 0.9,J = −0.003, λˆ = 0.1 0.9,J = −0.003, λˆ = 0.1 steady state solutions mLCE 1.4 0.00 t 5000 10 000 15 000 20 000 1.2 -0.02 1.0

0.8 -0.04

0.6 -0.06 0.4 -0.08 0.2

0.0 t 0 500 1000 1500 2000-0.10 (c) SIM and slow ﬂow for A˜ = 0.07, Bˆ =(d) maximal LCE for A˜ = 0.07, Bˆ = 0.044,J = −0.073, λˆ = 0.1 0.044,J = −0.073, λˆ = 0.1 steady state solutions mLCE 0.010 1.4 0.005 1.2

1.0 0.000 t 10 000 20 000 30 000 40 000 50 000 0.8 -0.005 0.6 -0.010 0.4

0.2 -0.015

0.0 t 0 500 1000 1500 2000-0.020 (e) SIM and slow ﬂow for A˜ = −0.083, Bˆ =(f) maximal LCE for A˜ = −0.083, Bˆ = −0.061,J = −0.07, λˆ = 0.1 −0.061,J = −0.07, λˆ = 0.1 Figure 4.2: Regular orbits

65 mLCE steady state solutions 0.02 1.4

1.2 0.01 1.0

0.8 0.00 t 2000 4000 6000 8000 10 000 0.6

0.4 -0.01 0.2

0.0 t 0 200 400 600 800 1000-0.02 (a) SIM and slow ﬂow for λˆ = 0.01 (b) maximal LCE for λˆ = 0.01

mLCE steady state solutions 0.00 t 1.4 1000 2000 3000 4000 1.2

1.0 -0.05

0.8 -0.10 0.6

0.4 -0.15 0.2

0.0 t 0 200 400 600 800 1000-0.20 (c) SIM and slow ﬂow for λˆ = 0.2 (d) maximal LCE for λˆ = 0.2 Figure 4.3: A˜ = −0.113, Bˆ = 0.124,J = 0.135, = 0.1, B¯ = −0.073 ω20

66 Relaxation Oscillations. They occur when there is bifurcations between all the branches, stable and unstable, of the SIM. In one case for a certain period of time only the upper branch exists. Then, after a saddle-node bifurcation, two other branches appear, the one stable and the other unstable. This phenomenon takes place for a period of time that depends in the parameters A,˜ B,ˆ λˆ and C.ˆ Then the branches of the SIM coalesce through saddle-node bifurcations. In this case the orbits of the slow flow are attracted near the stable branch of the SIM and make a sudden jump to the other stable branch when there is the bifurcation. Relaxation oscillations may exist also when there is bifurcations between two branches of the SIM (the one always unstable) while the other remain intact. This behavior occurs for values of λˆ near the values that change the structure of the SIM from the third case (relaxation oscillations) to the fourth. So, for these values of λˆ, although the structure of the SIM has changed, the slow flow still undergo relaxation oscillations (Figure 4.7). For the parameters A˜ = −0.113, Bˆ = 0.206,J = 0.135, = 0.1, B¯ = −0.073, the ω20 bifurcation diagram shows that there is a sudden change of the regular orbit of the slow flow for λˆ ≈ 0.4 (Figure 4.4). This sudden change of the regular orbits indicates the existence of relaxation oscillations. From Figures 4.5 we see that in fact, for λˆ < 0.4, the slow flow has relaxation oscillations. That means that the slow flow oscillates regularly in the region of the stable branch of the SIM until the time that saddle node bifurcation occurs and the the slow flow jumps to the region of the other, stable branch of the SIM, where it oscillates regularly.

Figure 4.4: bifurcation diagram for A˜ = −0.113, Bˆ = 0.206,J = 0.135, = 0.1, B¯ = −0.073 ω20

For the parameters A˜ = −0.093, Bˆ = −0.175,J = 0.105, = 0.1, B¯ = −0.055, the ω20 bifurcation diagram shows that the regular orbit has also a sudden change (Figure 4.6). As we see in Figures 4.7 for λˆ < 0.1, as we expect from the bifurcation diagram, the slow ﬂow performs relaxation oscillation. The slow ﬂow oscillates regularly and then there is

67 mLCE steady state solutions 1.4 0.00 t 500 1000 1500 2000 1.2

1.0 -0.05

0.8 -0.10 0.6

0.4 -0.15 0.2

0.0 t 0 500 1000 1500 2000-0.20 (a) SIM and slow ﬂow for λ = 0.4 (b) maximal LCE for λ = 0.4

mLCE steady state solutions 0.0 t 500 1000 1500 2000 1.4

1.2 -0.1 1.0

0.8 -0.2 0.6

0.4 -0.3 0.2

0.0 t 0 500 1000 1500 2000-0.4 (c) SIM and slow flow for λ = 0.45 (d) maximal LCE for λ = 0.45 Figure 4.5: A˜ = −0.113, Bˆ = 0.206,J = 0.135, = 0.1, B¯ = −0.073 ω20 the saddle node bifurcation of the SIM that leads to a sudden change in the amplitude of the oscillations. For λˆ = 0.1 the orbit of the slow flow oscillates near the stable branch of the SIM, and for certain initial conditions it may oscillate in the upper stable branch, then jump when we have the bifurcation, to the lower stable branch and stay there. This is the special case, that we mentioned above, where we see that even though the structure of the SIM is changed from the third case to the fourth, the slow flow still performs relaxation oscillations. This is due to the fact that the branches of the SIM are closed and the orbit may jump from the one basin of attraction to the other. We must indicate here, that the maximal LCEs are negative and there is only one point in the Poincare sections for different initial conditions, showing that the orbits of the slow flow are regular in the above cases of relaxation oscillations and are attracted to a periodic orbit. Many other examples of relaxation oscillations are given in Figure (4.8)

68 Figure 4.6: bifurcation diagram for A˜ = −0.093, Bˆ = −0.175,J = 0.105, = 0.1, B¯ = ω20 −0.055

mLCE steady state solutions 0.2 1.4

1.2 0.1 1.0

0.8 0.0 t 500 1000 1500 2000 0.6

0.4 -0.1 0.2

0.0 t 0 500 1000 1500 2000-0.2 (a) SIM and slow ﬂow for λˆ = 0.09 (b) maximal LCE for λˆ = 0.09

mLCE steady state solutions 0.2 1.4

1.2 0.1 1.0

0.8 0.0 t 500 1000 1500 2000 0.6

0.4 -0.1 0.2

0.0 t 0 500 1000 1500 2000-0.2 (c) SIM and slow ﬂow for λˆ = 0.1 (d) maximal LCE for λˆ = 0.1 Figure 4.7: A˜ = −0.093, Bˆ = −0.175,J = 0.105, = 0.1, B¯ = −0.055 ω20

69 steady state solutions 1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 t 0 1000 2000 3000 4000 5000 6000 (a) bifurcation diagram for A˜ = 0.118, Bˆ =(b) SIM and slow ﬂow for A˜ = 0.118, Bˆ = 0.165,J = −0.095, = 0.1, B¯ = −0.026 0.165,J = −0.095, = 0.1, B¯ = ω20 ω20 −0.026, λˆ = 0.022

steady state solutions 1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 t 0 500 1000 1500 2000 (c) bifurcation diagram for A˜ = −0.05, Bˆ =(d) SIM and slow ﬂow for A˜ = −0.05, Bˆ = 0.296,J = 0.027, = 0.1, B¯ = −0.178 0.296,J = 0.027, = 0.1, B¯ = ω20 ω20 −0.178, λˆ = 0.32

steady state solutions 1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 t 0 500 1000 1500 2000 (e) bifurcation diagram for A˜ = −0.075, Bˆ =(f) SIM and slow ﬂow for A˜ = −0.075, Bˆ = −0.207,J = 0.082, = 0.1, B¯ = −0.107 −0.207,J = 0.082, = 0.1, B¯ = ω20 ω20 −0.107, λˆ = 0.13 Figure 4.8: Examples of relaxation oscillations

70 Chaotic Behavior. For different values of the parameters, A,˜ B,Jˆ and the damping parameter λˆ, there is chaotic behavior. We may have chaotic orbits in the case where the SIM has more than one stable branches that bifurcate. For the parameters A˜ = −0.111, Bˆ = 0.178,J = 0.109, = 0.1, B¯ = −0.335 the ω20 bifurcation diagram has also two regions with bifurcations (Figure 4.9), the first one for values of λˆ < 0.02 and the second for 0.06 < λˆ < 0.1. Also for λˆ < 0.3 there is a sudden change of the periodic orbit of the slow flow that indicates the existence of relaxation oscillations. In the first region, for λˆ = 0.01 the slow flow oscillates near the stable branch of the SIM with positive values of the maximal Lyapunov Characteristic Exponent and a chaotic attractor as given in Figure 4.10. Another example of chaotic behavior of the slow flow is given for the parameters A˜ = −0.127, Bˆ = 0.139,J = 0.125, = 0.1, B¯ = −0.328. The bifurcation diagram (Figure 4.11) ω20 indicates that there are also two regions of chaotic behavior. The first one is for values of the damping parameter λˆ < 0.005 and the second for 0.05 < λˆ < 0.07. Also for λˆ < 0.1 there is a sudden change and the slow flow performs relaxation oscillations Indeed, for λˆ = 0.059 the slow flow performs relaxation oscillations and chaotic behavior with a chaotic attractor as given in Figure 4.12.

4.2 Conclusions

The slow flow of a three degree of freedom dissipative system of linear coupled oscillators with an essentially nonlinear attachment has rich dynamics. There are cases when the slow flow oscillates near the region of the SIM regularly and is attracted to a periodic orbit. When the SIM has saddle node bifurcations the slow flow has relaxation oscillation. It is interesting to note that when we have bifurcations of the SIM, in one hand, the system may have regular behavior and in the other hand, for different values of the parameters, may have chaotic behavior and therefore the slow flow becomes unpredictable.

71 (a) λˆ ∈ (0, 0.6)

(b) λˆ ∈ (0, 0.1)

72 steady state solutions 1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 t 0 500 1000 1500 2000 (a) SIM and slow ﬂow

mLCE 0.020

0.015

0.010

0.005

0.000 t 10 000 20 000 30 000 40 000 50 000

-0.005

-0.010 (b) maximal LCE

px -0.5

-0.6

-0.7

-0.8

-0.9

-1.0

-1.1

x -0.6 -0.4 -0.2 0.2 0.4 0.6 (c) Poincare section Figure 4.10: A˜ = −0.111, Bˆ = 0.178,J = 0.109, = 0.1, B¯ = −0.335, λˆ = 0.01 ω20

73 (a) λˆ ∈ (0, 0.6)

(b) λˆ ∈ (0, 0.1) Figure 4.11: bifurcation diagram for A˜ = −0.127, Bˆ = 0.139,J = 0.125, = 0.1, B¯ = ω20 −0.328

74 steady state solutions 1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0 t 0 500 1000 1500 2000 (a) SIM and slow ﬂow

mLCE 0.010

0.005

0.000 t 10 000 20 000 30 000 40 000 50 000

-0.005

-0.010

-0.015

-0.020 (b) maximal LCE

0.7

0.6

0.5

0.4

0.3

0.2

0.1

x -0.4 -0.2 0.2 0.4 0.6 (c) Poincare section Figure 4.12: A˜ = −0.127, Bˆ = 0.139,J = 0.125, = 0.1, B¯ = −0.328, λˆ = 0.059 ω20

Chapter 5

The effect of the Slow Invariant Manifold and the Slow Flow on the energy transfer and dissipation of the initial system

As it is known from the literature [15, 13, 63], energy transfer and dissipation in a system of coupled linear and nonlinear oscillators is directly connected with the bifurcations of the SIM and the dynamics of the slow flow. In the previous chapters we showed that the slow flow of the system has rich dynamics, we also made a classification of the different cases of the SIM and defined analytically the conditions for each case. In this chapter we study the effect of the Slow Invariant Manifold and the dynamics of the slow flow of the system on the energy transfer from the linear to the nonlinear oscillator and the dissipation of the total energy of the initial system. From our numerical results we confirm that the SIM, its bifurcations and the dynamics of the slow flow, play an essential role in the energy transfer and dissipation of the system. The chapter is organized as follows. In the first section we calculate the energy of the system. In the second section we present numerical simulations for each case of the SIM and the behavior of the system’s energy transfer and dissipation. Finally we conclude in the third section.

5.1 The energy of the system

In order to study the effect of the SIM and the slow ﬂow on the energy transfer and dissipation of our initial system we will study the rate of the total energy dissipation of the system. In order to do so, we use as a tool the graph of the logarithm of the total energy versus time for the different cases of the SIM.

77 Furthermore, in order to study the possible energy transfer from the linear to the nonlinear oscillators we calculate the instantaneous energy that is stored in the nonlinear oscillator. The total energy of the system is calculated by the Hamiltonian of the initial system (3.1) without the dissipative terms and it is given by

1 C d a ener(t) = (y˙2 +x ˙ 2 +x ˙ 2) + (y − x )4 + (x − x )2 + x2. (5.1) 2 0 1 4 0 2 0 1 2 1 The energy that is stored in the nonlinear oscillator is given, as a fraction of the total energy, by 2 C 4 y˙ + (y − x0) nlnE(t) = 2 4 .100%. (5.2) 1 2 2 2 C 4 d 2 a 2 2 (y˙ +x ˙ 0 +x ˙ 1) + 4 (y − x0) + 2 (x0 − x1) + 2 x1

5.2 Numerical Simulations

As we discussed in the second chapter, the damping parameter λˆ plays an essential role in the bifurcations of the SIM and the dynamics of the slow ﬂow. The SIM may have one branch that is always stable, three branches, two of them stable and the one that lies between them unstable, or it may bifurcate. In what follows we present numerical simulations for different sets of parameters and initial conditions.

The structure of the SIM depends on the parameters a, d, and the initial conditions x˙ 0(0), x˙ 1(0) of the initial system (3.1), since the parameters A,˜ Bˆ and J are related to the parameters of the initial system through the relations (3.6), (3.8),(3.10) and (3.15). For every set of the parameters a, d there are different structures of the SIM that are related to the different initial conditions x˙ 0(0), x˙ 1(0) and affects the dynamics of the initial system. First, we study the cases where the SIM has no bifurcations and the slow flow has regular orbits, i.e. when the SIM has always one branch or when the SIM has always three branches. In Figures (5.1)-(5.3) we present some examples where the SIM has only one stable branch. In these cases the slow flow follows the SIM. The total energy of the system dissipates smoothly and, as we observe i n the diagrams of the instantaneous energy that is stored in the nonlinear oscillator, there is no energy transfer from the linear to the nonlinear oscillator. An important remark here is that for the cases where the slow flow oscillates rapidly before it follows the SIM (Figures 5.4-5.5), we detect energy transfer to the nonlinear oscillator of the system in the time interval before the slow flow follows the SIM. Furthermore, in these cases, the total energy of the system dissipates faster for the time interval when the slow flow oscillates rapidly around the SIM, than the period of time when the slow flow follows the SIM. The same behavior, as in the case where the SIM has only one stable branch, is seen when the SIM has always three branches (Figure 5.6). The amount of energy that is transfered from the linear to the nonlinear oscillator depends on the initial conditions, i.e. the initial energy that is given to the system.

78 Steady state

0.35

0.30

0.25

0.20

0.15

0.10

0.05

t 200 400 600 800 1000 (a) SIM and slow ﬂow

nlnEHtL 100

0 t 0 200 400 600 800 1000 (b) Instantaneous energy (%) stored in the nonlinear oscillator

lnHenerHtLL -5.2

-5.4

-5.6

-5.8

-6.0

-6.2

t 200 400 600 800 1000

79 Steady state 0.04

0.03

0.02

0.01

t 200 400 600 800 1000 (a) SIM and slow ﬂow

nlnEHtL 100

0 t 0 200 400 600 800 1000 (b) Instantaneous energy (%) stored in the nonlinear oscillator

lnHenerHtLL

-9.5

-10.0

-10.5

-11.0

-11.5

-12.0

t 200 400 600 800 1000 -13.0 (c) Total energy ˆ Figure 5.2: One stable branch of the SIM: a = 0.1, d = 0.5, λ = 0.3, x0 = 0.01, x1 = 0.01

80 Steady state 0.04

0.03

0.02

0.01

t 200 400 600 800 1000 (a) SIM and slow ﬂow

nlnEHtL 100

0 t 0 200 400 600 800 1000 (b) Instantaneous energy (%) stored in the nonlinear oscillator

lnHenerHtLL

-10

-11

-12

-13

-14

t 200 400 600 800 1000 (c) Total energy ˆ Figure 5.3: One stable branch of the SIM: a = 0.2, d = 0.2, λ = 0.5, x0 = 0.01, x1 = 0.01

81 Steady state 0.4

0.3

0.2

0.1

t 200 400 600 800 1000 (a) SIM and slow ﬂow

nlnEHtL 100

0 t 0 200 400 600 800 1000 (b) Instantaneous energy (%) stored in the nonlinear oscillator

lnHenerHtLL

-2.1

-2.2

-2.3

t 200 400 600 800 1000

82 Steady state

0.04

0.03

0.02

0.01

t 200 400 600 800 1000 (a) SIM and slow ﬂow

nlnEHtL 100

0 t 0 200 400 600 800 1000 (b) Instantaneous energy (%) stored in the nonlinear oscillator

lnHenerHtLL

-9.5

-10.0

-10.5

-11.0

-11.5

-12.0

t 200 400 600 800 1000

83 Steady state

0.8

0.6

0.4

0.2

t 100 200 300 400 500 (a) SIM and slow ﬂow

nlnEHtL 100

0 t 0 100 200 300 400 500 (b) Instantaneous energy (%) stored in the nonlinear oscillator

lnHenerHtLL -5.24

-5.26

-5.28

-5.30

-5.32

t 100 200 300 400 500

84 Steady state

3.5

3.0

2.5

2.0

1.5

1.0

0.5

t 200 400 600 800 (a) SIM and slow ﬂow

nlnEHtL 100

0 t 0 200 400 600 800 (b) Instantaneous energy (%) stored in the nonlinear oscillator

lnHenerHtLL

-2

-4

-6

-8

t 200 400 600 800 (c) Total energy ˆ Figure 5.7: Relaxation oscillations of the slow ﬂow: a = 4, d = 4, λ = 0.15, x0 = 0.9, x1 = 0.5

The next case in our study is where the SIM bifurcates. As we observe in Figures (5.7-5.9), in this case, energy is transferred to the nonlinear oscillator, irrespectively whether the slow flow performs relaxation oscillations or not. The rate of the total energy dissipation depends on whether energy is transferred to the nonlinear oscillator or not. Specifically, the systems disipates energy faster in the time intervals when there is energy transfer to the nonlinear oscillator. We must indicate here that the behavior of the initial system, in the case where the slow flow oscillates rapidly until it follows the SIM (with no bifurcations), differs from its behavior when

85 Steady state

3.5

3.0

2.5

2.0

1.5

1.0

0.5

t 200 400 600 800 1000 (a) SIM and slow ﬂow

nlnEHtL 100

0 t 0 200 400 600 800 1000 (b) Instantaneous energy (%) stored in the nonlinear oscillator

lnHenerHtLL

-2

-4

-6

t 200 400 600 800 1000

86 Steady state

3.5

3.0

2.5

2.0

1.5

1.0

0.5

t 200 400 600 800 (a) SIM and slow ﬂow

nlnEHtL 100

0 t 0 200 400 600 800 (b) Instantaneous energy (%) stored in the nonlinear oscillator

lnHenerHtLL

-2

-4

-6

-8

-10

-12

t 200 400 600 800 (c) Total energy ˆ Figure 5.9: The SIM bifurcates: a = 6, d = 6, λ = 0.15, x0 = 0.9, x1 = 0.5

87 4

t 200 400 600 800 1000

-2

-4

ˆ (a) a = 0.01, d = 1, λ = 0.1, x0 = 0.9, x1 = 0.5.

0.010

0.005

0.000 t 200 400 600 800 1000 1200 1400

-0.005

-0.010 ˆ (b) a = 0.2, d = 0.2, λ = 0.05, x0 = 0.01, x1 = 0.01. Figure 5.10: Oscillations of the initial system when the SIM has no bifurcations. Black line: y(t), thick gray line: x0(t), dashed gray line: x1(t) there are bifurcations. In the ﬁrst case energy transfer from the linear to the nonlinear oscillator occurs in the beginning of the oscillations, until the time when the system oscillates in a regular way (Figure 5.10). In the second case energy transfer from the linear to the nonlinear oscillator occurs at a time when the SIM bifurcates. After the energy transfer, the system dissipates its energy and the oscillations fade out (Figure 5.11).

88 t 600 800 1000 1200 1400

ˆ (a) a = 4, d = 4, λ = 0.15, x0 = 0.9, x1 = 0.5.

0.5

t 200 400 600 800

-0.5

ˆ (b) a = 6, d = 6, λ = 0.15, x0 = 0.9, x1 = 0.5. Figure 5.11: Oscillations of the initial system when the SIM bifurcates. Black line: y(t), thick gray line: x0(t), dashed gray line: x1(t)

5.3 Conclusions

From the above study we conclude that the bifurcations of the SIM and the dynamics of the slow ﬂow play an important role in the energy transfer from the linear to the nonlinear oscillator and the dissipation of the total energy of the initial system. When the SIM has no bifurcations, there is no energy transfer from the linear to the nonlinear oscillator and the energy dissipates smoothly. When the SIM has bifurcations, then energy transfer occurs. Furthermore, when there is energy transfer to the nonlinear oscillator, the rate of the dissipation of the total energy of the system becomes larger. When the slow ﬂow oscillates rapidly around the SIM, the energy is transfered to the nonlinear oscillator. The amount of energy that transfers is related to the initial energy given to the system. The damping parameter λˆ determines whether the SIM bifurcates, that is, in order for the SIM to bifurcate, the damping parameter must satisfy the relation λˆ < √1 . Therefore, from the 3 above analysis, we conclude that the damping parameter determines the ability of the system to transfer energy from the linear to the nonlinear oscillator and plays a role in the rate of the total energy dissipation of the system.

Chapter 6

A nonlinear electric circuit for experimental implementation of the system

In the last decades, research activities in systems of nonlinear oscillators resulted in a lot of publications on phenomena that such systems exhibit [6, 7]. Also, the interesting dynamical behavior, which these systems have shown, has triggered an investigation in possible applications of such systems in various scientific fields, such as secure communications [58], cryptography [72], broadband communication systems [11] random number generators [73, 74], radars [32], robots [75] and in variety of complex physical, chemical and biological systems [57]. Such systems can be easily implemented by nonlinear electrical circuits [38, 76]. The first who did this was Professor Leon Chua in 1983. At that time, there was a deep desire to implement nonlinear circuits that allow the experimental demonstration of various phenomena, especially chaos, in order to refuse the claim that these phenomena were only a mathematical invention. This led Chua to investigate the possibility of designing an autonomous circuit behaving in a chaotic way [8]. So, this approach helps us to make experiments testing the dynamical behavior of nonlinear oscillators. In this Chapter we present a nonlinear electrical circuit that implements the reduced system (3.9). In more details, the reduced system is a Duffing-type nonlinear oscillator, which is driven by two sinusoidal voltage sources with different frequencies. We have made the simulations of the electrical circuit with the use of the MultiSim platform and we have also solved numerically the nonlinear system of differential equations using programming languages, such as Mathematica and TrueBasic. Furthermore, the estimation of maximal Lyapunov Exponent and the Poincare sections help us to identify the dynamical behavior of the above system. Finally, various types of oscillation such as periodic, quasiperiodic, and chaotic have been shown. The chapter is organized as follows. In the first section, the proposed nonlinear system and the electric circuit, which realizes the system, are presented. The dynamical behavior of the

91 proposed system and the simulation results are presented in Section two. Finally, the conclusion remarks are presented in Section three.

6.1 The Nonlinear System and the Proposed Electric Circuit

The reduced system (3.9) is described by the following set of differential equations:

dx = y, dt dy = −λy − Cx3 + A sin(2πf τ) + B sin(2πf τ), (6.1) dt 11 22 where x ≡ w of the reduced system (3.9), λ is the damping parameter, C is the coefﬁcient of the nonlinear term, A, B are the amplitudes of the external sinusoidal excitations, 2πf11 =

ω1, 2πf22 = ω2 and f11, f22 their frequencies. We must notice here, that although there are no limits in the values of the frequencies, interesting dynamical phenomena occur, when the ratio of the frequencies is irrational. The circuit topology that was adopted in the MultiSim platform in order to realize the system’s equations (6.1), is presented in Figure (6.1).

The proposed circuit consists of two identical operational ampliﬁers U1, U2(LF 411), and two multipliers U3,U4(AD734AN). Also, it should be mentioned that the signals x and y represent the voltages at the outputs of the operational ampliﬁers U1 and U2, while U3 and U4 realize the cubic term Cx3. Finally, the DC voltages used for all the ICs were V ± = ±15V .

In the system’s equations (6.1), parameters λ, C, A, B, f11 and f22 are deﬁned as follows:

R1 R1 R1 λ = ,C = ,A = V1, 2R5 200R2 2R3 R1 R1c R1c B = V2, f11 = f1, f22 = f2. (6.2) 2R4 2 2

1 R1c Also, the normalized time (τ) is τ = α t, whereα = 2 is the time constant of the circuit. The circuit’s elements values are: R1 = 10kΩ and c = 10nF , while the rest of the components

(R2,R3,R4,R5,V1,V2, f11 and f22) can be varied, in order to have a complete view of the dependence of the circuit on the values of the system’s parameters.

6.2 Simulation Results

We have studied the above circuit of Figure (6.1) using the Multisim platform and programming languages such as Mathematica and TrueBasic. The parameter C was ﬁxed to be C = 0.2 , while the amplitudes A, B and the frequencies f11, f22 of the voltage sources respectively, can be varied.

92 Figure 6.1: The proposed circuit emulating the nonlinear system.

In the case of the system stimulated by only one voltage source, U0 sin(2πft), the bifurcation diagrams x vs.U0, are shown in Figure 6.2, for λ = 0.10 and f = 0.07, f = 0.13 respectively. In the case f11 = f22 = f, the system follows the dynamics of the single source, where U0 = A+B.

93 (a) f = 0.07

(b) f = 0.13

Figure 6.2: Bifurcation diagrams x vs. U0, for λ = 0.1.

In the case A = 4 and B = 0, only the ﬁrst voltage source is activated, and the phase portrait of the system is shown in Figure (6.3). In Figure (6.3-a), the simulated phase portrait is shown, while in Figure (6.3-b), the phase portrait is shown obtained by using the Multisim platform. In the case A = 0 and B = 4, only the second voltage source is activated, and the phase portraits of the system are shown in Figure (6.4).

94 (a) By using programming language

(b) By using the Multisim platform

Figure 6.3: Phase portraits for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 4 and B = 0.

In the case A = 4 and B = 4, both voltage sources are activated, and the phase portraits of the system are shown in Figure (6.5). The system’s dynamics is chaotic with positive value of the maximal Lyapunov Characteristic Exponent equals to 0.0281. The Poincare section of the system is shown in Figure (6.6) exhibiting multi-band chaos. In the case A = 7,B = 15, The phase portraits y vs. x are shown in Figure (6.7), and the Poincare sections in Figure (6.8). In Figure (6.8-a), the surface of section is determined by the relation Mod(γ1(t), 2π) = 0, while in Figure (6.8-b) by the relation Mod(γ2(t), 2π) = 0. The maximal Lyapunov Characteristic Exponent is equal to 0.0350. As the value of the damping parameter λ is increased, the complexity of the system, when only one voltage source is activated, is decreased, as it is shown in Figures (6.9) and (6.10). A similar behavior is observed, when the

95 (a) By using programming language

(b) By using the Multisim platform

Figure 6.4: Phase portraits for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 0 and B = 4. frequency of the source is increased, for low values of λ, as it is shown in Figure (6.10-b).

96 (a) By using programming language

(b) By using the Multisim platform

Figure 6.5: Phase portraits for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 4 and B = 4.

97 Figure 6.6: Poincare section for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 4 and B = 4. Multiband chaos

98 (a) By using programming language

(b) By using the Multisim platform

Figure 6.7: Phase portraits for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 7 and B = 15.

99 (a) The surface of section is determined by the relation Mod(γ1(t), 2π) = 0

(b) The surface of section is determined by the relation Mod(γ2(t), 2π) = 0

Figure 6.8: Poincare section for λ = 0.1, f11 = 0.07 and f22 = 0.13 in the case A = 7 and B = 15. Strange Attractors

100 (a) λ = 0.1, f = 0.17

(b) λ = 0.35, f = 0.17

Figure 6.9: Bifurcation diagrams, x vs. U0, when only one voltage source is activated

101 (a) λ = 0.45, f = 0.17

(b) λ = 0.1, f = 0.41 Figure 6.10: For higher values of the damping factor λ, or for higher values of the frequency, period-1 oscillations are observed, as the value of the amplitude of the voltage source is increased. Bifurcation diagrams x vs. U0.

102 Quasiperiodicity

For λ = 0.10,C = 0.2, f11 = 0.70 and f22 = 1.835 the system is in a period-1 state for all values of amplitude, when only one voltage source is activated. When both sources are activated, the system’s behavior is changed. In Figure (6.11), the phase portraits are shown for A = −20.2 and B = 82.3. In this case high periodicity is observed. The negative sign of A means a phase difference of 1800.

(a) By using programming language

(b) By using the Multisim platform

Figure 6.11: Phase portraits for λ = 0.1, f11 = 0.7 and f22 = 1.835, A = −20.2, B = 82.3 and C = 0.2.

For A = 20.2,B = 82.3, f11 = 0.70, f22 = 1.835,C = 0.2 and λ = 0.10, the phase portraits are shown in Figure (6.12), while the Poincare section is shown in Figure (6.13). The close curve indicates quasiperiodic behavior of one-torus attractor. If we increase the value of the

103 damping factor, i.e. for A = 20.2,B = 82.3, f11 = 0.70, f22 = 1.835,C = 0.2 and λ = 0.15, the phase portraits of the system are shown in Figure (6.14), while the Poincare section is shown in Figure (6.15). The three closed curves indicate quasiperiodic behavior, but in this case we have a three-torus attractor. To the best of our knowledge, a three-tori attractor has been observed only by Manimehan et al. [47] in a modiﬁed canonical Chua’s circuit.

(a) By using programming language

(b) By using the Multisim platform

Figure 6.12: Phase portraits of y vs.x for λ = 0.1, f11 = 0.7 and f22 = 1.835, A = 20.2, B = 82.3 and C = 0.2.

104 Figure 6.13: Poincare section for λ = 0.1, f11 = 0.7 and f22 = 1.835 in the case A = 20.2 and B = 82.3 and C = 0.2. A case of quasiperiodicity.

6.3 Conclusions

In the present chapter, we have studied a Duffing-type circuit driven by two sinusoidal voltage sources having different frequencies. The state equations of the system were simulated by an electric nonlinear circuit, in order to use the Multisim platform, and compare the results of this real time platform with the results of the numerical simulation by using programming languages (Mathematica and TrueBasic). The simulations confirmed the rich dynamics of the above nonlinear system. Depending on the system parameters, periodic, quasiperiodic and chaotic oscillations were observed. Important role in the dynamics of the system plays the damping parameter λ, and the frequencies f11 and f22 of the external sinusoidal voltage sources. The simultaneous excitation of the circuit by two different input signals gives rise to unpredictable dynamics and the response of the system, i.e. the output signal, can be used as a random generator. The quasiperiodic behavior and the observation of a 3-tori attractor is quite a new dynamics for a Duffing-type system.

105 (a) By using programming language

(b) By using the Multisim platform

Figure 6.14: Phase portraits of y vs.x for λ = 0.15, f11 = 0.7 and f22 = 1.835, A = 20.2, B = 82.3 and C = 0.2.

106 Figure 6.15: Poincare section for λ = 0.15, f11 = 0.7 and f22 = 1.835 in the case A = 20.2 and B = 82.3 and C = 0.2. A paradigm of 3-tori quasiperiodicity.

107

Chapter 7

Epilogue and suggestions for further work

Linear and nonlinear coupled oscillators may transfer energy between them. This phenomenon can be very important in a wide variety of mechanical, electronic and other applications. In our work we studied a dissipative system composed of two coupled linear oscillators and a nonlinear oscillator that interacts through an essential nonlinearity with one of the linear oscillators. The mass of the nonlinear attachment is small in comparison to the masses of the linear oscillators and therefore the problem is singular. The Slow Invariant Manifolds and the slow flow of the system play important role on the dynamics of the system and affect the energy transfer from the linear to the nonlinear oscillator and the rate of the total energy dissipation. 0 0 Depending on the parameters and the initial conditions (x0(0), x1 (0)) of the system, we have shown through dynamical analysis and Tikhonov’s theorem, that the SIM can either have one branch that is stable or three branches, two branches stable and a third unstable branch that lies between them, or it may bifurcate. The damping parameter λˆ is essential for the behavior of the SIM. For values of λˆ > √1 the 3 SIM has always one stable branch. For values of of λˆ < √1 the SIM may have always three 3 branches or bifurcate. According to Tichonov’s theorem the orbits of the slow flow tend to these stable branches of the SIM for the interval of time that they exist. The structure of the SIM may be classified in five cases. The first case is when the SIM has a single branch, there are no bifurcations and the orbits of the slow flow tend to the stable branch of the SIM. The second case is when the SIM has three branches, two of them are stable and one between them unstable. On this case there are also no bifurcations. The SIM possesses three branches and for different initial conditions the orbits of the slow flow either tend to the upper or to the lower stable branch of the SIM.

109 The third case is when relaxation oscillations occur, with the slow flow to make transitions between the two stable branches of the SIM. In this case bifurcations occur as the orbits of the dynamical system undergo transitions between all branches of the SIM. For a certain period of the slow time scale only the upper branch exists. Then, after a saddle-node bifurcation, two additional branches appear, the one stable and the other unstable. This phenomenon takes place for a period of the slow time scale that depends on the parameters A,˜ B,ˆ λˆ and C.ˆ Then the stable upper branch and unstable branch of the SIM coalesce through saddle-node bifurcations. Therefore, the orbits of the slow flow that are initially attracted to the upper branch of the SIM, make a sudden transition (jump) to the lower branch producing relaxation oscillations. After sufficient increase of the slow time scale the upper branch reappears and the lower one disappears again through a saddle node bifurcation. This behavior is repeated periodically and gives rise to sustained relaxation oscillations in the dynamics. The fourth case is when bifurcations can occur either, between the upper stable branch of the SIM and the unstable branch, or only between the lower stable branch of the SIM and the unstable branch, resulting into the disappearance of a pair of stable-unstable branches of the SIM for a certain period of the slow time scale. The orbit of the slow flow is captured in the persisting stable branch of the SIM or in extreme cases, for values of the damping parameter λˆ that change the structure of the SIM from the third case to the fourth case, the slow flow may still perform relaxation oscillations. Finally, the fifth case corresponds to the excitation of periodic or non periodic orbits for certain initial conditions. For sufficiently small values of damping λˆ the orbits of the slow flow may oscillate rapidly, while approaching the stable branch of the SIM, and follow it until the time where there is bifurcations between the stable and the unstable branches of the SIM. Then an orbit is excited and the entire phenomenon is repeated. As we discussed earlier, when bifurcations occur as the orbits of the dynamical system undergo transitions between all branches of the SIM, the slow flow makes relaxation oscillations. We note that, in most of the cases, when relaxation oscillations occur, the maximal LCEs are negative and there is only one point in the Poincare sections for different initial conditions, showing that the orbits of the slow flow are regular and are attracted to a periodic orbit. For different values of the parameters there is chaotic behavior. This may occur for different values of the damping parameter λˆ for which the SIM has more than one stable branches that bifurcate. When there is chaotic behavior, the slow flow oscillates rapidly near the branches of the SIM. Energy transfer from the linear to the nonlinear oscillator and the rate of the total energy dissipation is related to the SIM, its bifurcation, and the dynamics of the slow flow. When the SIM has no bifurcations and the slow flow has regular orbits, that is, when the SIM has always one branch or when the SIM has always three branches, there is no energy transfer from the linear to the nonlinear oscillator. In the contrary, when the slow flow oscillates rapidly before it follows the SIM, we detect

110 energy transfer to the nonlinear oscillator of the system for the time interval before the slow flow follows the SIM. Furthermore, on these cases, the total energy of the system dissipates faster for the time interval that the slow flow oscillates rapidly around the SIM, than that for which the slow flow follows the SIM. When the SIM bifurcates, eventhough the slow flow performs relaxation oscillation or not, the system transfers energy to the nonlinear oscillator. The rate of the total energy dissipation differs from a time interval to another. Specifically, the systems disipates energy faster when there is energy transfer to the nonlinear oscillator. In our work we proposed a nonlinear electrical circuit that implements the reduced system (3.9). Our aim was to propose an experimental setup where we could investigate the different behaviors of the system. The reduced system is a Duffing-type nonlinear oscillator, which is driven by two sinusoidal voltage sources with different frequencies and has rich dynamics. Depending on the system parameters, periodic, quasiperiodic and chaotic oscillations were observed. As we expected by our analysis, and confirmed by the experimental study, important role in the dynamics of the system plays the damping parameter λ. Additionally, important role play the frequencies f11 and f22 of the external sinusoidal voltage sources. The simultaneous excitation of the circuit by two different input signals gives rise to unpredictable dynamics and the response of the system, i.e. the output signal, can be used as a random generator. The quasiperiodic behavior and the observation of a 3-tori attractor is quite a new dynamics for a Duffing-type system.

Suggestions for further work:

1. A more detailed study of the system for small values of the damping parameter λˆ 0.1.

2. The relation between the chaotic behavior and the energy transfer and dissipation of the system.

3. Study the behavior of the system for the subharmonic resonances (1:3) and (1:5).

4. Study similar systems with non essential nonlinearities.

5. Study similar systems with higher order nonlinearity, compare the different behaviors of the SIM, the slow ﬂow and the energy transfer and dissipation.

6. Further investigation of the nonlinear electric circuit and possible applications on electronic devices.

7. Proposal of a nonlinear electric circuit that implements the initial system.

111

Appendix A

AD734AN

113 10 MHz, 4-Quadrant a Multiplier/Divider AD734

FEATURES CONNECTION DIAGRAM High Accuracy 14-Lead DIP 0.1% Typical Error (Q Package and N Package) High Speed 10 MHz Full-Power Bandwidth 450 V/␮s Slew Rate X1 1 14 VP POSITIVE SUPPLY X INPUT 200 ns Settling to 0.1% at Full Power X2 2 13 DD DENOMINATOR DISABLE Low Distortion U0 3 12 W OUTPUT AD734 DENOMINATOR –80 dBc from Any Input U1 4 11 Z1 INTERFACE TOP VIEW (Not to Scale) Z INPUT Third-Order IMD Typically –75 dBc at 10 MHz U2 5 10 Z2 Low Noise Y1 6 9 ER REFERENCE VOLTAGE 94 dB SNR, 10 Hz to 20 kHz Y INPUT Y2 7 8 VN NEGATIVE SUPPLY 70 dB SNR, 10 Hz to 10 MHz Direct Division Mode 2 MHz BW at Gain of 100 APPLICATIONS High Performance Replacement for AD534 Multiply, Divide, Square, Square Root Modulator, Demodulator Wideband Gain Control, RMS-DC Conversion demodulator with input frequencies as high as 40 MHz as long Voltage-Controlled Amplifiers, Oscillators, and Filters as the desired output frequency is less than 10 MHz. Demodulator with 40 MHz Input Bandwidth The AD734AQ and AD734BQ are specified for the industrial ° ° PRODUCT DESCRIPTION temperature range of –40 C to +85 C and come in a 14-lead The AD734 is an accurate high speed, four-quadrant analog ceramic DIP. The AD734SQ/883B, available processed to ° ° multiplier that is pin-compatible with the industry-standard MIL-STD-883B for the military range of –55 C to +125 C, is AD534 and provides the transfer function W = XY/U. The available in a 14-lead ceramic DIP. AD734 provides a low-impedance voltage output with a full- power (20 V pk-pk) bandwidth of 10 MHz. Total static error PRODUCT HIGHLIGHTS (scaling, offsets, and nonlinearities combined) is 0.1% of full The AD734 embodies more than two decades of experience in scale. Distortion is typically less than –80 dBc and guaranteed. the design and manufacture of analog multipliers, to provide: The low capacitance X, Y and Z inputs are fully differential. In 1. A new output amplifier design with more than twenty times µ µ most applications, no external components are required to the slew-rate of the AD534 (450 V/ s versus 20 V/ s) for a define the function. full power (20 V pk-pk) bandwidth of 10 MHz. The internal scaling (denominator) voltage U is 10 V, derived 2. Very low distortion, even at full power, through the use of from a buried-Zener voltage reference. A new feature provides circuit and trimming techniques that virtually eliminate all of the option of substituting an external denominator voltage, the spurious nonlinearities found in earlier designs. allowing the use of the AD734 as a two-quadrant divider with a 3. Direct control of the denominator, resulting in higher 1000:1 denominator range and a signal bandwidth that remains multiplier accuracy and a gain-bandwidth product at small 10 MHz to a gain of 20 dB, 2 MHz at a gain of 40 dB and denominator values that is typically 200 times greater than 200 kHz at a gain of 60 dB, for a gain-bandwidth product of that of the AD534 in divider modes. 200 MHz. 4. Very clean transient response, achieved through the use of a The advanced performance of the AD734 is achieved by a novel input stage design and wide-band output amplifier, combination of new circuit techniques, the use of a high speed which also ensure that distortion remains low even at high complementary bipolar process and a novel approach to laser- frequencies. trimming based on ac signals rather than the customary dc 5. Superior noise performance by careful choice of device methods. The wide bandwidth (>40 MHz) of the AD734’s geometries and operating conditions, which provide a input stages and the 200 MHz gain-bandwidth product of the guaranteed 88 dB of dynamic range in a 20 kHz bandwidth. multiplier core allow the AD734 to be used as a low distortion

REV. C Information furnished by Analog Devices is believed to be accurate and reliable. However, no responsibility is assumed by Analog Devices for its use, nor for any infringements of patents or other rights of third parties One Technology Way, P.O. Box 9106, Norwood, MA 02062-9106, U.S.A. which may result from its use. No license is granted by implication or Tel: 781/329-4700 World Wide Web Site: http://www.analog.com otherwise under any patent or patent rights of Analog Devices. Fax: 781/326-8703 © Analog Devices, Inc., 1999 ؇ ≥ ⍀ AD734–SPECIFICATIONS (TA = +25 C, +VS = VP = +15 V, –VS = VN = –15 V, RL 2 k ) TRANSFER FUNCTION   ()X1 − X2 ()Y1 −Y2  W = AO  − ()Z1 − Z2   ()U −U   1 2  ABS Parameter Conditions Min Typ Max Min Typ Max Min Typ Max Units MULTIPLIER PERFORMANCE Transfer Function W = XY/10 W = XY/10 W = XY/10 Total Static Error1 –10 V ≤ X, Y ≤ 10 V 0.1 0.4 0.1 0.25 0.1 0.4 % Over TMIN to TMAX 1 0.6 1.25 % vs. Temperature TMIN to TMAX 0.004 0.003 0.004 %/°C vs. Either Supply ±VS = 14 V to 16 V 0.01 0.05 0.01 0.05 0.01 0.05 %/V Peak Nonlinearity –10 V ≤ X ≤ +10 V, Y = +10 V 0.05 0.05 0.05 % –10 V ≤ Y ≤ +10 V, X = +10 V 0.025 0.025 0.025 % THD2 X = 7 V rms, Y = +10 V, f ≤ 5 kHz –58 –66 –58 dBc TMIN to TMAX –55 –63 –55 dBc Y = 7 V rms, X = +10 V, f ≤ 5 kHz –60 –80 –60 dBc TMIN to TMAX –57 –74 –57 dBc Feedthrough X = 7 V rms, Y = nulled, f ≤ 5 kHz –85 –60 –85 –70 –85 –60 dBc Y = 7 V rms, X = nulled, f ≤ 5 kHz –85 –66 –85 –76 –85 –66 dBc Noise (RTO) X = Y = 0 Spectral Density 100 Hz to 1 MHz 1.0 1.0 1.0 µV/√Hz Total Output Noise 10 Hz to 20 kHz –94 –88 –94 –88 –94 –88 dBc TMIN to TMAX –85 –85 –85 dBc DIVIDER PERFORMANCE (Y = 10 V) Transfer Function W = XY/U W = XY/U W = XY/U Gain Error Y = 10 V, U = 100 mV to 10 V 1 1 1 % X Input Clipping Level Y ≤ 10 V 1.25 × U 1.25 × U 1.25 × UV U Input Scaling Error3 0.3 0.15 0.3 % TMIN to TMAX 0.8 0.65 1 % (Output to 1%) U = 1 V to 10 V Step, X = 1 V 100 100 100 ns

INPUT INTERFACES (X, Y, & Z) 3 dB Bandwidth 40 40 40 MHz Operating Range Differential or Common Mode ±12.5 ±12.5 ±12.5 V X Input Offset Voltage 15 5 15 mV TMIN to TMAX 25 15 25 mV Y Input Offset Voltage 10 5 10 mV TMIN to TMAX 12 6 12 mV Z Input Offset Voltage 20 10 20 mV TMIN to TMAX 50 50 90 mV Z Input PSRR (Either Supply) f ≤ 1 kHz 54 70 66 70 54 70 dB TMIN to TMAX 50 56 50 dB CMRR f = 5 kHz 70 85 70 85 70 85 dB Input Bias Current (X, Y, Z Inputs) 50 300 50 150 50 300 nA TMIN to TMAX 400 300 500 nA Input Resistance Differential 50 50 50 kΩ Input Capacitance Differential 2 2 2 pF

DENOMINATOR INTERFACES (U0, U1, & U2) Operating Range VN to VP-3 VN to VP-3 VN to VP-3 V Denominator Range 1000:1 1000:1 1000:1 Interface Resistor U1 to U2 28 28 28 kΩ

OUTPUT AMPLIFIER (W) Output Voltage Swing TMIN to TMAX ±12 ±12 ±12 V Open-Loop Voltage Gain X = Y = 0, Input to Z 72 72 72 dB Dynamic Response From X or Y Input, CL ≤ 20 pF 3 dB Bandwidth W ≤ 7 V rms 8 10 8 10 8 10 MHz Slew Rate 450 450 450 V/µs Settling Time +20 V or –20 V Output Step To 1% 125 125 125 ns To 0.1% 200 200 200 ns Short-Circuit Current TMIN to TMAX 20 50 80 20 50 80 20 50 80 mA

POWER SUPPLIES, ±VS Operating Supply Range ±8 ±16.5 ±8 ±16.5 ±8 ±16.5 V Quiescent Current TMIN to TMAX 691269126912mA NOTES 1Figures given are percent of full scale (e.g., 0.01% = 1 mV). 2dBc refers to deciBels relative to the full-scale input (carrier) level of 7 V rms. 3See Figure 10 for test circuit. All min and max specifications are guaranteed. Specifications subject to change without notice. –2– REV. C AD734

ABSOLUTE MAXIMUM RATINGS1 ORDERING GUIDE Supply Voltage ...... ±18 V Internal Power Dissipation2 Temperature Package Package ° Model Range Description Option for TJ max = 175 C ...... 500 mW X, Y and Z Input Voltages ...... VN to VP AD734AN –40°C to +85°C Plastic DIP N-14 Output Short Circuit Duration ...... Indefinite AD734BN –40°C to +85°C Plastic DIP N-14 Storage Temperature Range AD734AQ –40°C to +85°C Cerdip Q-14 Q ...... –65°C to +150°C AD734BQ –40°C to +85°C Cerdip Q-14 Operating Temperature Range AD734SQ/883B –55°C to +125°C Cerdip Q-14 AD734A, B (Industrial) ...... –40°C to +85°C AD734SCHIPS –55°C to +125°CDie AD734S (Military) ...... –55°C to +125°C Lead Temperature Range (soldering 60 sec) ...... +300°C Transistor Count ...... 81 ESD Rating ...... 500 V NOTES 1Stresses above those listed under Absolute Maximum Ratings may cause perma- nent damage to the device. This is a stress rating only; functional operation of the device at these or any other conditions above those indicated in the operational section of this specification is not implied. 2 14-Lead Ceramic DIP: θJA = 110°C/W.

CHIP DIMENSIONS & BONDING DIAGRAM Dimensions shown in inches and (mm). (Contact factory for latest dimensions.)

REV. C –3– AD734

is typically less than 5 mV, which corresponds to a bias current X1 X = X1 – X2 HIGH-ACCURACY XIF TRANSLINER of only 100 nA. This low bias current ensures that mismatches X2 DD MULTIPLIER CORE in the sources resistances at a pair of inputs does not cause an offset error. These currents remain low over the full temperature U XZ XY/U – Z DENOMINATOR ∑ WIF W CONTROL U range and supply voltages. U0 A ∞ The common-mode range of the X, Y and Z inputs does not U1 ER O fully extend to the supply rails. Nevertheless, it is often possible Ru AD734 U2 to operate the AD734 with one terminal of an input pair con- Y1 Z1 nected to either the positive or negative supply, unlike previous YIF ZIF Y2 multipliers. The common-mode resistance is several megohms. Y = Y1 – Y2 Z = Z1 – Z 2 Z2 The full-scale output of ±10 V can be delivered to a load resis- Figure 1. AD734 Block Diagram tance of 1 kΩ (although the specifications apply to the standard Ω FUNCTIONAL DESCRIPTION multiplier load condition of 2 k ). The output amplifier is Figure 1 is a simplified block diagram of the AD734. Operation stable driving capacitive loads of at least 100 pF, when a slight is similar to that of the industry-standard AD534 and in many increase in bandwidth results from the peaking caused by this µ applications these parts are pin-compatible. The main functional capacitance. The 450 V/ s slew rate of the AD734’s output am- difference is the provision for direct control of the denominator plifier ensures that the bandwidth of 10 MHz can be maintained voltage, U, explained fully on the following page. Internal sig- up to the full output of 20 V pk-pk. Operation at reduced supply ± nals are actually in the form of currents, but the function of the voltages is possible, down to 8 V, with reduced signal levels. AD734 can be understood using voltages throughout, as shown Available Transfer Functions in this figure. Pins are named using upper-case characters (such The uncommitted (open-loop) transfer function of the AD734 is as X1, Z2) while the voltages on these pins are denoted by sub- scripted variables (for example, X , Z ). ()X − X ()Y − Y  1 2 = 1 2 1 2 − − W AO  ()Z1 Z2  , (1) The AD734’s differential X, Y and Z inputs are handled by  U  wideband interfaces that have low offset, low bias current and low distortion. The AD734 responds to the difference signals where AO is the open-loop gain of the output op-amp, typically X = X1 – X2, Y = Y1 – Y2 and Z = Z1 – Z2, and rejects 72 dB. When a negative feedback path is provided, the circuit common-mode voltages on these inputs. The X, Y and Z will force the quantity inside the brackets essentially to zero, interfaces provide a nominal full-scale (FS) voltage of ±10 V, resulting in the equation but, due to the special design of the input stages, the linear (X1 – X2)(Y1 – Y2) = U (Z1 – Z2) (2) range of the differential input can be as large as ±17 V. Also unlike previous designs, the response on these inputs is not This is the most useful generalized transfer function for the clipped abruptly above ±15 V, but drops to a slope of one half. AD734; it expresses a balance between the product XY and the product UZ. The absence of the output, W, in this equation The bipolar input signals X and Y are multiplied in a translinear only reflects the fact that we have not yet specified which of the core of novel design to generate the product XY/U. The inputs is to be connected to the op amp output. denominator voltage, U, is internally set to an accurate, temperature-stable value of 10 V, derived from a buried-Zener Most of the functions of the AD734 (including division, unlike reference. An uncalibrated fraction of the denominator voltage the AD534 in this respect) are realized with Z1 connected to W. U appears between the voltage reference pin (ER) and the So, substituting W in place of Z1 in the above equation results in negative supply pin (VN), for use in certain applications where an output. a temperature-compensated voltage reference is desirable. The ()X − X ()Y − Y internal denom-inator, U, can be disabled, by connecting the = 1 2 1 2 + W Z2. (3) denominator disable Pin 13 (DD) to the positive supply pin U (VP); the denom-inator can then be replaced by a fixed or The free input Z2 can be used to sum another signal to the variable external volt-age ranging from 10 mV to more than 10 V. output; in the absence of a product signal, W simply follows the The high-gain output op-amp nulls the difference between voltage at Z2 with the full 10 MHz bandwidth. When not XY/U and an additional signal Z, to generate the final output needed for summation, Z2 should be connected to the ground W. The actual transfer function can take on several forms, de- associated with the load circuit. We can show the allowable pending on the connections used. The AD734 can perform all polarities in the following shorthand form: of the functions supported by the AD534, and new functions using the direct-division mode provided by the U-interface. ()±X ()±Y ()±W = +±Z. ()+U (4) Each input pair (X1 and X2, Y1 and Y2, Z1 and Z2) has a differential input resistance of 50 kΩ; this is formed by “real” resistors (not a small-signal approximation) and is subject to a In the recommended direct divider mode, the Y input is set to a tolerance of ±20%. The common-mode input resistance is fixed voltage (typically 10 V) and U is varied directly; it may several megohms and the parasitic capacitance is about 2 pF. have any value from 10 mV to 10 V. The magnitude of the ratio X/U cannot exceed 1.25; for example, the peak X-input for U The bias currents associated with these inputs are nulled by = 1 V is ±1.25 V. Above this level, clipping occurs at the laser-trimming, such that when one input of a pair is optionally positive and negative extremities of the X-input. Alternatively, ac-coupled and the other is grounded, the residual offset voltage

–4– REV. C AD734

the AD734 can be operated using the standard (AD534) divider After temperature-correction (block TC), the reference voltage connections (Figure 8), when the negative feedback path is is applied to transistor Qd and trimmed resistor Rd, which established via the Y2 input. Substituting W for Y2 in Equation generate the required reference current. Transistor Qu and (2), we get resistor Ru are not involved in setting up the internal denominator, and their associated control pins U0, U1 and U2 will normally ()Z − Z be grounded. The reference voltage is also made available, via W = U 2 1 + Y . − 1 (5) the 100 kΩ resistor Rr, at Pin 9 (ER); the purpose of Qr is ()X1 X2 explained below. In this case, note that the variable X is now the denominator, When the control pin DD (denominator disable) is connected to and the above restriction (X/U ≤ 1.25) on the magnitude of the VP, the internal source of Iu is shut off, and the collector cur- X input does not apply. However, X must be positive in order rent of Qu must provide the denominator current. The resistor for the feedback polarity to be correct. Y can be used for 1 Ru is laser-trimmed such that the multiplier denominator is summing purposes or connected to the load ground if not exactly equal to the voltage across it (that is, across pins U1 and needed. The shorthand form in this case is U2). Note that this trimming only sets up the correct internal Ω ± ratio; the absolute value of Ru (nominally 28 k ) has a ()Z ± ()±W =+()U +±()Y . (6) tolerance of 20%. Also, the alpha of Qu, (typically 0.995) ()+X which might be seen as a source of scaling error, is canceled by the alpha of other transistors in the complete circuit. In some cases, feedback may be connected to two of the available inputs. This is true for the square-rooting connections In the simplest scheme (Figure 3), an externally-provided (Fig-ure 9), where W is connected to both X1 and Y2. Setting control voltage, VG, is applied directly to U0 and U2 and the X1 = W and Y2 = W in Equation (2), and anticipating the resulting voltage across Ru is therefore reduced by one VBE. For possibility of again providing a summing input, so setting X2 = S example, when VG = 2 V, the actual value of U will be about and Y1 = S, we find, in shorthand form 1.3 V. This error will not be important in some closed-loop applications, such as automatic gain control (AGC), but clearly ()±W =+()U ()+Z +±()S . (7) is not acceptable where the denominator value must be well- defined. When it is required to set up an accurate, fixed value of This is seen more generally to be the geometric-mean function, U, the on-chip reference may be used. The transistor Qr is since both U and Z can be variable; operation is restricted to provided to cancel the VBE of Qu, and is biased by an external one quadrant. Feedback may also be taken to the U-interface. resistor, R2, as shown in Figure 4. R1 is chosen to set the de- Full details of the operation in these modes is provided in the sired value of U and consists of a fixed and adjustable resistor. appropriate section of this data sheet. Direct Denominator Control A valuable new feature of the AD734 is the provision to replace VP 14 +VS AD734 the internal denominator voltage, U, with any value from +10 mV Iu ~60␮A U0 DD 13 to +10 V. This can be used (1) to simply alter the multiplier 3 Qu Rr scaling, thus improve accuracy and achieve reduced noise levels 100k⍀ when operating with small input signals; (2) to implement an U1 ER accurate two-quadrant divider, with a 1000:1 gain range and an VG NC 4 9 NC Ru Qr asymptotic gain-bandwidth product of 200 MHz; (3) to achieve 28k⍀ certain other special functions, such as AGC or rms. U2 VN 5 8 –VS Figure 2 shows the internal circuitry associated with denominator control. Note first that the denominator is actually proportional Figure 3. Low-Accuracy Denominator Control to a current, Iu, having a nominal value of 356 µA for U = 10 V, whereas the primary reference is a voltage, generated by a buried- Zener circuit and laser-trimmed to have a very low temperature Iu coefficient. This voltage is nominally 8 V with a tolerance of AD734 VP 14 +VS U0 ± Qu 10%. 3 Rr DD 13 100k⍀ R2 U1 ER NOMINALLY 14 VP 356␮A for 4 9 Iu U = 10V AD734 LINK TO Ru Qr DISABLE 28k⍀ NOM R1 U2 8V VN 13 DD NC 5 8 –VS Rr U0 3 Qu Qd 100k⍀

U1 4 TC 9 ER Figure 4. Connections for a Fixed Denominator Rd Ru Qr ⍀ NOM 28k 22.5k⍀ NOM Table I shows useful values of the external components for set- 8V U2 5 8 VN ting up nonstandard denominator values. NEGATIVE SUPPLY

Figure 2. Denominator Control Circuitry

REV. C –5– AD734

Table I. Component Values for Setting Up Nonstandard +15V AD734 Denominator Values X – INPUT 1 X1 VP 14 0.1␮F ؎ Denominator R1 (Fixed) R1 (Variable) R2 10V FS 2 X2 DD 13 NC (X1 – X2)(Y1 – Y2) 3 U0 W 12 W = + Z2 5 V 34.8 kΩ 20 kΩ 120 kΩ LOAD 10V 4 U1 Z1 11 L 3 V 64.9 kΩ 20 kΩ 220 kΩ GROUND 5 U2 Z2 10 Z2 2 V 86.6 kΩ 50 kΩ 300 kΩ OPTIONAL L Ω Ω Ω Y – INPUT 6 Y1 ER 9 NC 0.1␮F SUMMING INPUT ± ؎ V 174 k 100 k 620 k 1 10V FS 7 Y2 VN 8 10V FS The denominator can also be current controlled, by grounding –15V Pin 3 (U0) and withdrawing a current of Iu from Pin 4 (U1). Figure 5. Basic Multiplier Circuit The nominal scaling relationship is U = 28 × Iu, where u is expressed in volts and Iu is expressed in milliamps. Note, of 32 Hz. When a tighter control of this frequency is needed, or however, that while the linearity of this relationship is very good, when the HP corner is above about 100 kHz, an external resis- it is subject to a scale tolerance of ±20%. Note that the common tor should be added across the pair of input nodes. mode range on Pins 3 through 5 actually extends from 4 V to At least one of the two inputs of any pair must be provided with 36 V below VP, so it is not necessary to restrict the connection a dc path (usually to ground). The careful selection of ground of U0 to ground if it should be desirable to use some other returns is important in realizing the full accuracy of the AD734. voltage. The Z2 pin will normally be connected to the load ground, The output ER may also be buffered, re-scaled and used as a which may be remote, in some cases. It may also be used as an general-purpose reference voltage. It is generated with respect to optional summing input (see Equations (3) and (4), above) ± the negative supply line Pin 8 (VN), but this is acceptable when having a nominal FS input of 10 V and the full 10 MHz driving one of the signal interfaces. An example is shown in bandwidth. Figure 12, where a fixed numerator of 10 V is generated for a In applications where high absolute accuracy is essential, the divider application. There, Y2 is tied to VN but Y1 is 10 V above scaling error caused by the finite resistance of the signal source(s) this; therefore the common-mode voltage at this interface is still may be troublesome; for example, a 50 Ω source resistance at 5 V above VN, which satisfies the internal biasing requirements just one input will introduce a gain error of –0.1%; if both the (see Specifications table). X- and Y-inputs are driven from 50 Ω sources, the scaling error in the product will be –0.2%. Provided the source resistance(s) OPERATION AS A MULTIPLIER are known, this gain error can be completely compensated by All of the connection schemes used in this section are essentially including the appropriate resistance (50 Ω or 100 Ω, respectively, identical to those used for the AD534, with which the AD734 is in the above cases) between the output W (Pin 12) and the Z1 pin-compatible. The only precaution to be noted in this regard feedback input (Pin 11). If Rx is the total source resistance is that in the AD534, Pins 3, 5, 9, and 13 are not internally associated with the X1 and X2 inputs, and Ry is the total source connected and Pin 4 has a slightly different purpose. In many resistance associated with the Y1 and Y2 inputs, and neither Rx cases, an AD734 can be directly substituted for an AD534 with nor Ry exceeds 1 kΩ, a resistance of Rx+Ry in series with pin immediate benefits in static accuracy, distortion, feedthrough, Z1 will provide the required gain restoration. and speed. Where Pin 4 was used in an AD534 application to Pins 9 (ER) and 13 (DD) should be left unconnected in this achieve a reduced denominator voltage, this function can now be application. The U-inputs (Pins 3, 4 and 5) are shown much more precisely implemented with the AD734 using alter- connected to ground; they may alternatively be connected to native connections (see Direct Denominator Control, page 5). VN, if desired. In applications where Pin 2 (X2) happens to be Operation from supplies down to ±8 V is possible. The supply driven with a high-amplitude, high-frequency signal, the current is essentially independent of voltage. As is true of all capacitive coupling to the denominator control circuitry via an high speed circuits, careful power-supply decoupling is impor- ungrounded Pin 3 can cause high-frequency distortion. However, tant in maintaining stability under all conditions of use. The the AD734 can be operated without modification in an AD534 decoupling capacitors should always be connected to the load socket, and these three pins left unconnected, with the above ground, since the load current circulates in these capacitors at caution noted. high frequencies. Note the use of the special symbol (a triangle +15V with the letter ‘L’ inside it) to denote the load ground. AD734 0.1␮F 1 X – INPUT X1 VP 14 Standard Multiplier Connections ؎10V FS L 2 X2 DD 13 NC (X – X ) (Y – Y ) Figure 5 shows the basic connections for multiplication. The X 1 2 1 2 1 1 3 U0 W 12 I = + W R 50k⍀ and Y inputs are shown as optionally having their negative 10V S 4 Z1 11 R nodes grounded, but they are fully differential, and in many U1 S applications the grounded inputs may be reversed (to facilitate 5 U2 Z2 10 6 Y1 ER 9 NC interfacing with signals of a particular polarity, while achieving Y – INPUT 0.1␮F I ؎10mA MAX FS 10V FS W؎ some desired output polarity) or both may be driven. 7 Y2 VN 8 LOAD ؎10V MAXIMUM L L LOAD VOLTAGE The AD734 has an input resistance of 50 kΩ ± 20% at the X, –15V Y, and Z interfaces, which allows ac-coupling to be achieved Figure 6. Conversion of Output to a Current with moderately good control of the high-pass (HP) corner frequency; a capacitor of 0.1 µF provides a HP corner frequency

–6– REV. C AD734

Current Output R2 +15V It may occasionally be desirable to convert the output voltage to 1.6k⍀ AD734 0.1␮F 1 X1 VP 14 a current. In correlation applications, for example, multiplica- R1 L tion is followed by integration; if the output is in the form of a 1.6k⍀ 2 X2 DD 13 NC 3 U0 W 12 R3 current, a simple grounded capacitor can perform this function. 13k⍀ ω 2 Figure 6 shows how this can be achieved. The op amp forces Esin t 4 U1 Z1 11 E cos2ωt/10V R4 the voltage across Z1 and Z2, and thus across the resistor RS, to 5 U2 Z2 10 4.32k⍀ be the product XY/U. Note that the input resistance of the 9 6 Y1 ER NC 0.1␮F C L Z interface is in shunt with RS, which must be calculated 7 Y2 VN 8 accordingly. L –15V The smallest FS current is simply ±10 V/50 kΩ, or ±200 µA, Figure 7. Frequency Doubler with a tolerance of about 20%. To guarantee a 1% conversion Ω tolerance without adjustment, RS must be less than 2.5 k . The OPERATION AS A DIVIDER maximum full scale output current should be limited to about The AD734 supports two methods for performing analog ± Ω 10 mA (thus, RS = 1 k ). This concept can be applied to all division. The first is based on the use of a multiplier in a feed- connection modes, with the appropriate choice of terminals. back loop. This is the standard mode recommended for Squaring and Frequency-Doubling multipliers having a fixed scaling voltage, such as the AD534, Squaring of an input signal, E, is achieved simply by connecting and will be described in this Section. The second uses the the X and Y inputs in parallel; the phasing can be chosen to AD734’s unique capability for externally varying the scaling produce an output of E2/U or –E2/U as desired. The input may (denominator) voltage directly, and will be described in the have either polarity, but the basic output will either always be next section. positive or negative; as for multiplication, the Z2 input may be Feedback Divider Connections used to add a further signal to the output. Figure 8 shows the connections for the standard (AD534) When the input is a sine wave, a squarer behaves as a frequency- divider mode. Feedback from the output, W, is now taken to the doubler, since Y2 (inverting) input, which, provided that the X-input is positive, establishes a negative feedback path. Y1 should normally (Esinwt)2 = E2 (1 – cos2wt)/2 (8) be connected to the ground associated with the load circuit, but Equation (8) shows a dc term at the output which will vary may optionally be used to sum a further signal to the output. If strongly with the amplitude of the input, E. This dc term can be desired, the polarity of the Y-input connections can be reversed, avoided using the connection shown in Figure 7, where an with W connected to Y1 and Y2 used as the optional summation RC-network is used to generate two signals whose product has input. In this case, either the polarity of the X-input connections no dc term. The output is must be reversed, or the X-input voltage must be negative.

 E  π   E  π   1  +15V W = 4 sinwt +   sinwt −    (9) AD734 0.1␮F  4   4  10 V   2  2  X INPUT 1 X1 VP 14 +0.1V TO L for w = 1/CR1, which is just +10V 2 X2 DD 13 NC (Z2 – Z1) 2 3 U0 W 12 W = 10 +Y1 W = E (cos2wt)/( 10 V) (10) (X1 – X2) 4 U1 Z1 11 Z INPUT which has no dc component. To restore the output to ±10 V ؎10V FS 5 U2 Z2 10 when E = 10 V, a feedback attenuator with an approximate ratio Y 9 1 6 Y1 ER NC 0.1␮F of 4 is used between W and Z1; this technique can be used OPTIONAL L 8 SUMMING 7 Y2 VN wherever it is desired to achieve a higher overall gain in the L INPUT –15V transfer function. ؎10V FS In fact, the values of R3 and R4 include additional compensa- Figure 8. Standard (AD534) Divider Connection tion for the effects of the 50 kΩ input resistance of all three The numerator input, which is differential and can have either interfaces; R2 is included for a similar reason. These resistor polarity, is applied to pins Z1 and Z2. As with all dividers based values should not be altered without careful calculation of the on feedback, the bandwidth is directly proportional to the consequences; with the values shown, the center frequency f is 0 denominator, being 10 MHz for X = 10 V and reducing to 100 kHz for C = 1 nF. The amplitude of the output is only a 100 kHz for X = 100 mV. This reduction in bandwidth, and the weak function of frequency: the output amplitude will be 0.5% increase in output noise (which is inversely proportional to the too low at f = 0.9f and f = 1.1f . The cross-connection is simply 0 0 denominator voltage) preclude operation much below a denomi- to produce the cosine output with the sign shown in Equation nator of 100 mV. Division using direct control of the denominator (10); however, the sign in this case will rarely be important. (Figure 10) does not have these shortcomings.

REV. C –7– AD734

+15V This connection scheme may also be viewed as a variable-gain AD734 0.1␮F element, whose output, in response to a signal at the X input, is 1 X1 VP 14 L controllable by both the Y input (for attenuation, using Y less 2 X2 DD 13 NC D W =(10V) (Z2 – Z1)+ S than U) and the U input (for amplification, using U less than 3 U0 W 12 Y). The ac performance is shown in Figure 11; for these results, S 4 U1 Z1 11 Z INPUT Y was maintained at a constant 10 V. At U = 10 V, the gain is OPTIONAL +10mV TO SUMMING L 5 U2 Z2 10 +10V unity and the circuit bandwidth is a full 10 MHz. At U = 1 V, INPUT the gain is 20 dB and the bandwidth is essentially unaltered. At 10V FS 6 Y1 ER 9 NC ␮؎ 0.1 F U = 100 mV, the gain is 40 dB and the bandwidth is 2 MHz. 7 Y2 VN 8 L Finally, at U = 10 mV, the gain is 60 dB and the bandwidth is –15V 250 kHz, corresponding to a 250 MHz gain-bandwidth product.

Figure 9. Connection for Square Rooting 70 Connections for Square-Rooting U = 10mV The AD734 may be used to generate an output proportional to 60 the square-root of an input using the connections shown in 50

Figure 9. Feedback is now via both the X and Y inputs, and is U = 100mV always negative because of the reversed-polarity between these 40 two inputs. The Z input must have the polarity shown, but 30 because it is applied to a differential port, either polarity of GAIN – dB U = 1V input can be accepted with reversal of Z1 and Z2, if necessary. 20 The diode D, which can be any small-signal type (1N4148 being suitable) is included to prevent a latching condition which 10 U = 10V could occur if the input momentarily was of the incorrect 0 polarity of the input, the output will be always negative.

Note that the loading on the output side of the diode will be 10k 100k 1M 10M provided by the 25 kΩ of input resistance at X1 and Y2, and by FREQUENCY – Hz the user’s load. In high speed applications it may be beneficial Figure 11. Three-Variable Multiplier/Divider Performance to include further loading at the output (to 1 kΩ minimum) to The 2 MΩ resistor is included to improve the accuracy of the speed up response time. As in previous applications, a further gain for small denominator voltages. At high gains, the X input signal, shown here as S, may be summed to the output; if this offset voltage can cause a significant output offset voltage. To option is not used, this node should be connected to the load eliminate this problem, a low-pass feedback path can be used ground. from W to X2; see Figure 13 for details. DIVISION BY DIRECT DENOMINATOR CONTROL The AD734 may be used as an analog divider by directly vary- Where a numerator of 10 V is needed, to implement a two- ing the denominator voltage. In addition to providing much quadrant divider with fixed scaling, the connections shown in higher accuracy and bandwidth, this mode also provides greater Figure 12 may be used. The reference voltage output appearing flexibility, because all inputs remain available. Figure 10 shows between Pin 9 (ER) and Pin 8 (VN) is amplified and buffered the connections for the general case of a three-input multiplier by the second op amp, to impose 10 V across the Y1/Y2 input. divider, providing the function Note that Y2 is connected to the negative supply in this application. This is permissible because the common-mode voltage is ()X − X ()Y − Y still high enough to meet the internal requirements. The transfer W = 1 2 1 2 + Z ()U −U 2, (11) function is 1 2 where the X, Y, and Z signals may all be positive or negative,  −  = X1 X2 + W 10 V   Z2. but the difference U = U1 – U2 must be positive and in the − (12)  U1 U2  range +10 mV to +10 V. If a negative denominator voltage must be used, simply ground the noninverting input of the op amp. The ac performance of this circuit remains as shown in Figure 11.

As previously noted, the X input must have a magnitude of less +15V than 1.25U. AD734 1 +15V X1 VP 14 AD734 X – INPUT ␮ 2 X2 DD 13 0.1 F (X1– X2) 10V 1 X1 VP 14 U W = + Z 1 U –U 2 X – INPUT 0.1␮F 3 U0 W 12 1 2 DD 2 X2 13 (X1 – X2) (Y1 – Y2) W = + Z U – INPUT 2M⍀ LOAD U1 2 4 U1 Z1 11 3 U0 W 12 U1 – U2 L GROUND U – INPUT 2M⍀ LOAD U 5 U2 Z2 10 Z 4 U1 Z1 11 L GROUND 2 2 OPTIONAL 6 Y1 ER 9 ␮ L U2 5 U2 Z2 10 Z2 0.1 F SUMMING OPTIONAL INPUT ؎ L 7 Y2 VN 8 6 Y1 ER 9 NC 0.1␮F SUMMING INPUT 10V FS Y – INPUT 100k⍀ 10V FS ⍀؎ 7 Y2 VN 8 200k SCALE –15V ADJUST –15V OP AMP = AD712 DUAL Figure 10. Three-Variable Multiplier/Divider Using Direct Denominator Control Figure 12. Two-Quadrant Divider with Fixed 10 V Scaling

–8– REV. C AD734

A PRECISION AGC LOOP The output amplitude tracks EC over the range +1 V to slightly The variable denominator of the AD734 and its high gain- more than +10 V. bandwidth product make it an excellent choice for precise automatic gain control (AGC) applications. Figure 13 shows a +2 suggested method. The input signal, EIN, which may have a peak amplitude of from 10 mV to 10 V at any frequency from +1 100 Hz to 10 MHz, is applied to the X input, and a fixed posi- 100kHZ tive voltage EC to the Y input. Op amp A2 and capacitor C2 dB

form an integrator having a current summing node at its invert- – ing input. (The AD712 dual op amp is a suitable choice for this 0 application.) In the absence of an input, the current in D2 and R2 causes the integrator output to ramp negative, clamped by ERROR diode D3, which is included to reduce the time required for the –1 loop to establish a stable, calibrated, output level once the 100HZ 1MHZ circuit has received an input signal. With no input to the denominator (U0 and U2), the gain of the AD734 is very high –2 (about 70 dB), and thus even a small input causes a substantial 10m 100m 1 10 output. INPUT AMPLITUDE – Volts Figure 14. AGC Amplifier Output Error vs. Input Voltage R3 ⍀ C1 A1 1M 1␮F AD734 +15V WIDEBAND RMS-DC CONVERTER 1 X1 VP 14 USING U INTERFACE EIN 0.1␮F 2 X2 DD 13 D1 The AD734 is well suited to such applications as implicit RMS- 1N914 DC conversion, where the AD734 implements the function 3 U0 W 12 EOUT D3 C2 NC 4 U1 Z1 11 C1 1N914 1␮F A2 L 1␮F 2 avg[] V IN 5 U2 Z2 10 V = (13) 0.1␮F L RMS EC 6 Y1 ER 9 V RMS +1V TO D2 7 Y2 VN 8 +10V 1N914 using its direct divide mode. Figure 15 shows the circuit. –15V +15V R2 R1 ⍀ ⍀ OP AMP = AD712 DUAL 1M 1M AD734 0.1␮F VIN 1 X1 VP 14 Figure 13. Precision AGC Loop 1/2 AD708 L 2 X2 DD 13 R1 U2b L 3.32k⍀ Diode D1 and C1 form a peak detector, which rectifies the out- V 3 U0 W 12 O put and causes the integrator to ramp positive. When the U2a U1 1/2 4 U1 Z1 11 C1 C2 AD708 current in R1 balances the current in R2, the integrator output 47␮F 1␮F 5 U2 Z2 10 holds the denominator output at a constant value. This occurs L L V = V 2 L O IN when there is sufficient gain to raise the amplitude of E to that 6 Y1 ER 9 IN 0.1␮F required to establish an output amplitude of EC over the range 7 Y2 VN 8 L L of +1 V to +10 V. The X input of the AD734, which has finite –15V offset voltage, could be troublesome at the output at high gains. The output offset is reduced to that of the X input (one or two Figure 15. A 2-Chip, Wideband RMS-DC Converter millivolts) by the offset loop comprising R3, C3, and buffer A1. In this application, the AD734 and an AD708 dual op amp The low pass corner frequency of 0.16 Hz is transformed to a serve as a 2-chip RMS-DC converter with a 10 MHz bandwidth. high-pass corner that is multiplied by the gain (for example, Figure 16 shows the circuit’s performance for square-, sine-, 160 Hz at a gain of 1000). and triangle-wave inputs. The circuit accepts signals as high as In applications not requiring operation down to low frequencies, 10 V p-p with a crest factor of 1 or 1 V p-p with a crest factor of amplifier A1 can he eliminated, but the AD734’s input 10. The circuit’s response is flat to 10 MHz with an input of resistance of 50 kΩ between X1 and X2 will reduce the time 10 V, flat to almost 5 MHz for an input of 1 V, and to almost constant and increase the input offset. Using a non-polar 20 µF 1 MHz for inputs of 100 mV. For accurate measurements of tantalum capacitor for C1 will result in the same unity-gain input levels below 100 mV, the AD734’s output offset (Z inter- high-pass corner; in this case, the offset gain increases to 20, still face) voltage, which contributes a dc error, must be trimmed out. very acceptable. In Figure 15’s circuit, the AD734 squares the input signal, and 2 Figure 14 shows the error in the output for sinusoidal inputs at its output (VIN ) is averaged by a low-pass filter that consists of R1 and C1 and has a corner frequency of 1 Hz. Because of the 100 Hz, 100 kHz, and 1 MHz, with EC set to +10 V. The output error for any frequency between 300 Hz and 300 kHz is implicit feedback loop, this value is both the output value, VRMS, similar to that for 100 kHz. At low signal frequencies and low and the denominator in Equation (13). U2a and U2b, an input amplitudes, the dynamics of the control loop determine AD708 dual dc precision op amp, serve as unity-gain buffers, the gain error and distortion; at high frequencies, the 200 MHz supplying both the output voltage and driving the U interface. gain-bandwidth product of the AD734 limit the available gain.

REV. C –9– AD734

100 If the two X1 inputs are at frequencies f1 and f2 and the frequency at the Y1 input is f0, then the two-tone third-order 10 intermodulation products should appear at frequencies 2f1 – f2 ± f0 and 2f2 – f1 ± f0. Figures 18 and 19 show the output spectra of the AD734 with f = 9.95 MHz, f = 10.05 MHz, and f = 1 1 2 0 9.00 MHz for a signal level of f1 & f2 of 6 dBm and f0 of +24 dBm in Figure 18 and f1 & f2 of 0 dBm and f0 of +24 dBm 100m in Figure 19. This performance is without external trimming of the AD734’s X and Y input-offset voltages. 10m The possible Two Tone Intermodulation Products are at 2 × OUTPUT VOLTAGE – Volts SQUARE WAVE 9.95 MHz – 10.05 MHz ± 9.00 MHz and 2 × 10.05 – 1m SINE WAVE 9.95 MHz ± 9.00 MHz; of these only the third-order products TRI-WAVE at 0.850 MHz and 1.150 MHz are within the 10 MHz band-

100␮ width of the AD734; the desired output signals are at 10k 100k 1M 10M 0.950 MHz and 1.050 MHz. Note that the difference (Figure INPUT FREQUENCY – Hz 18) between the desired outputs and third-order products is Figure 16. RMS-DC Converter Performance approximately 78 dB, which corresponds to a computed third-order intercept point of +46 dBm. LOW DISTORTION MIXER The AD734’s low noise and distortion make it especially suitable for use as a mixer, modulator, or demodulator. Although the AD734’s –3 dB bandwidth is typically 10 MHz and is established by the output amplifier, the bandwidth of its X and Y interfaces and the multiplier core are typically in excess of 40 MHz. Thus, provided that the desired output signal is less than 10 MHz, as would typically be the case in demodulation, the AD734 can be used with both its X and Y input signals as high as 40 MHz. One test of mixer performance is to linearly combine two closely spaced, equal-amplitude sinusoidal signals and then mix them with a third signal to determine the mixer’s 2-tone Third-Order Intermodulation Products.

+15V HP3326A AD734 0.1␮F COMBINE 1 14 A + B X1 VP AD707 2 X2 DD 13 DATEL HP3585A DVC-8500 3 U0 W 12 WITH 10X PROBE Figure 18. AD734 Third-Order Intermodulation Performance dBm REF TO 50⍀ for f = 9.95 MHz, f = 10.05 MHz, and f = 9.00 MHz and for 4 U1 Z1 11 2k⍀ 1 2 0 Signal Levels of f1 & f2 of 6 dBm and f0 of +24 dBm. All Dis- 5 U2 Z2 10 HP3326A played Signal Levels Are Attenuated 20 dB by the 10X Probe HIGH VOLTAGE 9 6 Y1 ER 0.1␮F OPTION Used to Measure the Mixer’s Output 7 Y2 VN 8 –15V Figure 17. AD734 Mixer Test Circuit Figure 17 shows a test circuit for measuring the AD734’s performance in this regard. In this test, two signals, at 10.05 MHz and 9.95 MHz are summed and applied to the AD734’s X interface. A second 9 MHz signal is applied to the AD734’s Y interface. The voltage at the U interface is set to 2 V to use the full dynamic range of the AD734. That is, by connecting the W and Z1 pins together, grounding the Y2 and X2 pins, and setting U = 2 V, the overall transfer function is

X1Y1 W = (14) 2 V and W can be as high as 20 V p-p when X1 = 2 V p-p and Y1 = 10 V p-p. The 2 V p-p signal level corresponds to +10 dBm into a 50 Ω input termination resistor connected from X1 or Y1 to Figure 19. AD734 Third-Order Intermodulation Performance ground. for f1 = 9.95 MHz, f2 = 10.05 MHz, and f0 = 9.00 MHz and for Signal Levels of f1 & f2 of 0 dBm and f0 of +24 dBm. All Dis- played Signal Levels Are Attenuated 20 dB by the 10X Probe Used to Measure the Mixer’s Output –10– REV. C Typical Characteristics–AD734

؎ VS = 15V ؎ ؎ VS = 15V VS = 15V X = 1.4V RMS 0.06 0.3 R = 2k⍀ R = 2k⍀ Y = 10V LOAD LOAD ⍀ RLOAD = 500 0.04 CLOAD = 20pF CLOAD = 20pF 0.2 CLOAD = 20pF 0.02 0.1 0.1 0 0.05 0

–0.02 0 –0.1

–0.04 –0.05 GAIN FLATNESS –0.2

DIFFERENTIAL GAIN – dB –0.06 –0.1 –0.3

–0.08 DIFFERENTIAL PHASE – Degrees –0.4

–2V0 2V –2V0 2V 100k 1M 10M SIGNAL AMPLITUDE SIGNAL AMPLITUDE FREQUENCY – Hz Figure 20. Differential Gain at Figure 21. Differential Phase at Figure 22. Gain Flatness, 300 kHz to 3.58 MHz and RL = 2 kΩ 3.58 MHz and RL = 2 kΩ 10 MHz, RL = 500 Ω

100 100 0 INPUT SIGNAL = 7V RMS Y INPUT, X = 10V 80 80 Ð40

Y INPUT, X NULLED

60 60 Ð60 VN

dB X INPUT, Y NULLED X INPUT, Y = 10V VP Ð80 40 PSRR Ð dB 40

COMMON-MODE FEEDTHROUGH Ð dBc 20 20 Ð100 SIGNAL = 7V RMS

0 0 1k 10k100k 1M 10M 1k 10k100k 1M 10M 1k 10k100k 1M 10M FREQUENCY Ð Hz FREQUENCY Ð Hz FREQUENCY Ð Hz

Figure 23. CMRR vs. Frequency Figure 24. PSRR vs. Frequency Figure 25. Feedthrough vs. Frequency

0 0 0

TEST INPUT = 1V RMS U = 2V TEST INPUT = 7V RMS –20 –20 –20 FREQUENCY = 1MHz OTHER INPUT = 2V DC OTHER INPUT = 10V DC VP = 15V Ն ⍀ RLOAD = 2k VN = –15V ⍀ RLOAD = 2k –40 –40 –40 dBc dBc X INPUT dBc X INPUT X INPUT. Y = 10V DC –60 –60 –60 Y INPUT Y INPUT Y INPUT. X = 10V DC –80 –80 –80

–100 1k 10k100k 1M 10M 1k 10k100k 1M 10M –10dBm 10dBm 30dBm FREQUENCY – Hz FREQUENCY – Hz 70.7mV RMS 707mV RMS 7V RMS SIGNAL LEVEL Figure 26. THD vs. Frequency, Figure 27. THD vs. Frequency, Figure 28. THD vs. Signal Level, U = 2 V U = 10 V f = 1 MHz

REV. C –11– AD734–Typical Characteristics 5 ؎ VS = 15V 4 X = 1.4V rms Y = 10V 0 3 ⍀ RLOAD = 500 –30 2 CLOAD = 20pF, 47pF, 100pF INCREASING ؎ C –60 1 LOAD VS = 15V 0 –90 X = 1.4V rms Y = 10V –120 ⍀ –1 RLOAD = 500 CLOAD = 20pF, 47pF, 100pF AMPLITUDE – dB –2 –150 PHASE SHIFT – Degrees –3 –180 INCREASING –4 CLOAD –5 100k 1M 10M 100k 1M 10M Figure 31. Pulse Response vs. C1442a–0–10/99 FREQUENCY – Hz FREQUENCY – Hz Figure 29. Gain vs. Frequency vs. Figure 30. Phase vs. Frequency CLOAD, CLOAD = 0 pF, 47 pF, 100 pF, 200 pF CLOAD vs. CLOAD 0 20

10 Ð10 U = 1V U = 2V U = 5V U = 10V 5

0 Ð20 –5

X1 FREQ = OUTPUT AMPLITUDE Ð dB Y FREQ Ð1MHz OUTPUT SWING – Volts –10 1 (E.G., Y1 Ð X1 = 1MHz –15 FOR ALL CURVES) Ð30 1020 30 40 5060 70 80 90 100 –20 Y1 FREQUENCY Ð MHz 818910111213 14151617 ؎ SUPPLY VOLTAGE – VS Figure 32. Output Swing vs. Supply Figure 33. Output Amplitude vs. Input Voltage Frequency, When Used as Demodulator

20 60 8 INPUT OFFSET VOLTAGE INPUT OFFSET VOLTAGE INPUT OFFSET VOLTAGE 15 DRIFT WILL TYPICALLY BE 40 DRIFT WILL TYPICALLY BE 6 DRIFT WILL TYPICALLY BE WITHIN SHADED AREA WITHIN SHADED AREA WITHIN SHADED AREA 10 20 4

5 0 2

0 –20 0

DEVIATION OF INPUT –5 DEVIATION OF INPUT –2 DEVIATION OF INPUT –40 OFFSET VOLTAGE – mV OFFSET VOLTAGE – mV OFFSET VOLTAGE – mV

–10 –60 –4

–15 –6 –55 –35 –15 5 25 45 65 85 105 125 –55 –35 –15 5 25 45 65 85 105 125 –55 –35 –15 5 25 45 65 85 105 125 TEMPERATURE – ؇C TEMPERATURE – ؇C TEMPERATURE – ؇C Figure 34. VOS Drift, X Input Figure 35. VOS Drift, Z Input Figure 36. VOS Drift, Y Input OUTLINE DIMENSIONS Dimensions shown in inches and (mm).

14-Lead Plastic DIP (N) Package 14-Lead Ceramic DIP (Q) Package

14 8

0.280 (7.11) PRINTED IN U.S.A. PIN 1 0.240 (6.10) 1 7

0.795 (20.19) 0.325 (8.25) 0.725 (18.42) 0.300 (7.62) 0.060 (1.52) 0.195 (4.95) 0.210 0.015 (0.38) (5.33) 0.115 (2.93) MAX 0.130 0.160 (4.06) (3.30) 0.015 (0.381) 0.115 (2.93) MIN 0.008 (0.204)

0.022 (0.558) 0.100 0.070 (1.77) SEATING 0.014 (0.356) (2.54) 0.045 (1.15) PLANE BSC –12– REV. C

Appendix B

LF411

127 LF411-N www.ti.com SNOSBH6D –APRIL 1998–REVISED MARCH 2013 LF411 Low Offset, Low Drift JFET Input Operational Amplifier Check for Samples: LF411-N

1FEATURES DESCRIPTION These devices are low cost, high speed, JFET input 23• Internally trimmed offset voltage: 0.5 mV(max) operational amplifiers with very low input offset • Input offset voltage drift: 10 μV/°C(max) voltage and specified input offset voltage drift. They • Low input bias current: 50 pA require low supply current yet maintain a large gain • Low input noise current: 0.01 pA/√Hz bandwidth product and fast slew rate. In addition, well matched high voltage JFET input devices provide • Wide gain bandwidth: 3 MHz(min) very low input bias and offset currents. The LF411 is • High slew rate: 10V/μs(min) pin compatible with the standard LM741 allowing • Low supply current: 1.8 mA designers to immediately upgrade the overall • High input impedance: 1012Ω performance of existing designs. • Low total harmonic distortion: ≤0.02% These amplifiers may be used in applications such as high speed integrators, fast D/A converters, sample • Low 1/f noise corner: 50 Hz and hold circuits and many other circuits requiring low • Fast settling time to 0.01%: 2 μs input offset voltage and drift, low input bias current, high input impedance, high slew rate and wide bandwidth.

Typical Connection

Figure 1.

Connection Diagram

Note: Pin 4 connected to case. Figure 2. TO – Top View See Package Number NEV0008A

1 Please be aware that an important notice concerning availability, standard warranty, and use in critical applications of Texas Instruments semiconductor products and disclaimers thereto appears at the end of this data sheet.

2BI-FET II is a trademark of dcl_owner.

3All other trademarks are the property of their respective owners.

PRODUCTION DATA information is current as of publication date. Copyright © 1998–2013, Texas Instruments Incorporated Products conform to specifications per the terms of the Texas Instruments standard warranty. Production processing does not necessarily include testing of all parameters. LF411-N

SNOSBH6D –APRIL 1998–REVISED MARCH 2013 www.ti.com

Figure 3. PDIP – Top View See Package Number P0008E

These devices have limited built-in ESD protection. The leads should be shorted together or the device placed in conductive foam during storage or handling to prevent electrostatic damage to the MOS gates.

Absolute Maximum Ratings(1) LF411A LF411 Supply Voltage ±22V ±18V Differential Input Voltage(2) ±38V ±30V ±19V ±15V Output Short Circuit Duration Continuous Continuous

TO Package PDIP Package Power Dissipation(3) (4) 670 mW 670 mW

Tjmax 150°C 115°C

θjA 162°C/W (Still Air) 120°C/W 65°C/W (400 LF/min Air Flow)

θjC 20°C/W Operating Temp. Range See (5) See (5)

Storage Temp. Range −65°C≤TA≤150°C −65°C≤TA≤150°C Lead Temp. (Soldering, 10 sec.) 260°C 260°C ESD Tolerance Rating to be determined.

(1) “Absolute Maximum Ratings” indicate limits beyond which damage to the device may occur. Operating Ratings indicate conditions for which the device is functional, but do not ensure specific performance limits. (2) Unless otherwise specified the absolute maximum negative input voltage is equal to the negative power supply voltage. (3) For operating at elevated temperature, these devices must be derated based on a thermal resistance of θjA. (4) Max. Power Dissipation is defined by the package characteristics. Operating the part near the Max. Power Dissipation may cause the part to operate outside specified limits. (5) These devices are available in both the commercial temperature range 0°C≤TA≤70°C and the military temperature range −55°C≤TA≤125°C. The temperature range is designated by the position just before the package type in the device number. A “C” indicates the commercial temperature range and an “M” indicates the military temperature range. The military temperature range is available in the TO package only.

2 Submit Documentation Feedback Copyright © 1998–2013, Texas Instruments Incorporated Product Folder Links: LF411-N LF411-N www.ti.com SNOSBH6D –APRIL 1998–REVISED MARCH 2013 DC Electrical Characteristics (1)(2) LF411A LF411 Symbol Parameter Conditions Units Min Typ Max Min Typ Max

VOS Input Offset Voltage RS=10 kΩ, TA=25°C 0.3 0.5 0.8 2.0 mV ΔV /ΔT Average TC of Input R =10 kΩ (3) 20 (3) OS S 7 10 7 μV/°C Offset Voltage (2) (4) IOS Input Offset Current VS=±15V Tj=25°C 25 100 25 100 pA

Tj=70°C 2 2 nA

Tj=125°C 25 25 nA (2) (4) IB Input Bias Current VS=±15V Tj=25°C 50 200 50 200 pA

Tj=70°C 4 4 nA

Tj=125°C 50 50 nA 12 12 RIN Input Resistance Tj=25°C 10 10 Ω

AVOL Large Signal Voltage VS=±15V, VO=±10V, 50 200 25 200 V/mV

Gain RL=2k, TA=25°C Over Temperature 25 200 15 200 V/mV

VO Output Voltage Swing VS=±15V, RL=10k ±12 ±13.5 ±12 ±13.5 V

VCM Input Common-Mode ±16 +19.5 ±11 +14.5 V Voltage Range −16.5 −11.5 V CMRR Common-Mode Rejection R ≤10k S 80 100 70 100 dB Ratio PSRR Supply Voltage Rejection See (5) 80 100 70 100 dB Ratio

IS Supply Current 1.8 2.8 1.8 3.4 mA (1) RETS 411X for LF411MH and LF411MJ military specifications. (2) Unless otherwise specified, the specifications apply over the full temperature range and for VS=±20V for the LF411A and for VS=±15V for the LF411. VOS, IB, and IOS are measured at VCM=0. (3) The LF411A is 100% tested to this specification. The LF411 is sample tested to insure at least 90% of the units meet this specification. (4) The input bias currents are junction leakage currents which approximately double for every 10°C increase in the junction temperature, Tj. Due to limited production test time, the input bias currents measured are correlated to junction temperature. In normal operation the junction temperature rises above the ambient temperature as a result of internal power dissipation, PD. Tj=TA+θjA PD where θjA is the thermal resistance from junction to ambient. Use of a heat sink is recommended if input bias current is to be kept to a minimum. (5) Supply voltage rejection ratio is measured for both supply magnitudes increasing or decreasing simultaneously in accordance with common practice, from ±15V to ±5V for the LF411 and from ±20V to ±5V for the LF411A.

AC Electrical Characteristic (1)(2) LF411A LF411 Symbol Parameter Conditions Units Min Typ Max Min Typ Max

SR Slew Rate VS=±15V, TA=25°C 10 15 8 15 V/μs

GBW Gain-Bandwidth Product VS=±15V, TA=25°C 3 4 2.7 4 MHz e Equivalent Input Noise Voltage T =25°C, R =100Ω, n A S 25 25 nV / √Hz f=1 kHz

in Equivalent Input Noise Current TA=25°C, f=1 kHz 0.01 0.01 pA / √Hz

THD Total Harmonic Distortion AV=+10, RL=10k, VO=20 <0.02 <0.02 % Vp-p, BW=20 Hz−20 kHz

(1) Unless otherwise specified, the specifications apply over the full temperature range and for VS=±20V for the LF411A and for VS=±15V for the LF411. VOS, IB, and IOS are measured at VCM=0. (2) RETS 411X for LF411MH and LF411MJ military specifications.

SNOSBH6D –APRIL 1998–REVISED MARCH 2013 www.ti.com Typical Performance Characteristics Input Bias Current Input Bias Current

Figure 4. Figure 5.

Positive Common-Mode Supply Current Input Voltage Limit

Figure 6. Figure 7.

Negative Common-Mode Input Voltage Limit Positive Current Limit

Figure 8. Figure 9.

4 Submit Documentation Feedback Copyright © 1998–2013, Texas Instruments Incorporated Product Folder Links: LF411-N LF411-N www.ti.com SNOSBH6D –APRIL 1998–REVISED MARCH 2013 Typical Performance Characteristics (continued) Negative Current Limit Output Voltage Swing

Figure 10. Figure 11.

Output Voltage Swing Gain Bandwidth

Figure 12. Figure 13.

Bode Plot Slew Rate

Figure 14. Figure 15.

SNOSBH6D –APRIL 1998–REVISED MARCH 2013 www.ti.com Typical Performance Characteristics (continued) Distortion vs Undistorted Output Frequency Voltage Swing

Figure 16. Figure 17.

Open Loop Frequency Common-Mode Rejection Response Ratio

Figure 18. Figure 19.

Power Supply Equivalent Input Noise Rejection Ratio Voltage

Figure 20. Figure 21.

6 Submit Documentation Feedback Copyright © 1998–2013, Texas Instruments Incorporated Product Folder Links: LF411-N LF411-N www.ti.com SNOSBH6D –APRIL 1998–REVISED MARCH 2013 Typical Performance Characteristics (continued) Open Loop Voltage Gain Output Impedance

Figure 22. Figure 23.

Inverter Settling Time

Figure 24.

PULSE RESPONSE (RL=2 KΩ, CL10 PF)

Figure 25. Small Signal Inverting Figure 26. Small Signal Non-Inverting

SNOSBH6D –APRIL 1998–REVISED MARCH 2013 www.ti.com

Figure 27. Large Signal Inverting Figure 28. Large Signal Non-Inverting

Figure 29. Current Limit (RL=100Ω) APPLICATION HINTS

The LF411 series of internally trimmed JFET input op amps (BI-FET II™ ) provide very low input offset voltage and specified input offset voltage drift. These JFETs have large reverse breakdown voltages from gate to source and drain eliminating the need for clamps across the inputs. Therefore, large differential input voltages can easily be accommodated without a large increase in input current. The maximum differential input voltage is independent of the supply voltages. However, neither of the input voltages should be allowed to exceed the negative supply as this will cause large currents to flow which can result in a destroyed unit. BI-FET II™ Exceeding the negative common-mode limit on either input will force the output to a high state, potentially causing a reversal of phase to the output. Exceeding the negative common-mode limit on both inputs will force the amplifier output to a high state. In neither case does a latch occur since raising the input back within the common-mode range again puts the input stage and thus the amplifier in a normal operating mode. Exceeding the positive common-mode limit on a single input will not change the phase of the output; however, if both inputs exceed the limit, the output of the amplifier may be forced to a high state. The amplifier will operate with a common-mode input voltage equal to the positive supply; however, the gain bandwidth and slew rate may be decreased in this condition. When the negative common-mode voltage swings to within 3V of the negative supply, an increase in input offset voltage may occur.

The LF411 is biased by a zener reference which allows normal circuit operation on ±4.5V power supplies. Supply voltages less than these may result in lower gain bandwidth and slew rate. The LF411 will drive a 2 kΩ load resistance to ±10V over the full temperature range. If the amplifier is forced to drive heavier load currents, however, an increase in input offset voltage may occur on the negative voltage swing and finally reach an active current limit on both positive and negative swings. Precautions should be taken to ensure that the power supply for the integrated circuit never becomes reversed in polarity or that the unit is not inadvertently installed backwards in a socket as an unlimited current surge through the resulting forward diode within the IC could cause fusing of the internal conductors and result in a destroyed unit. As with most amplifiers, care should be taken with lead dress, component placement and supply decoupling in order to ensure stability. For example, resistors from the output to an input should be placed with the body close to the input to minimize “pick-up” and maximize the frequency of the feedback pole by minimizing the capacitance from the input to ground. A feedback pole is created when the feedback around any amplifier is resistive. The parallel resistance and capacitance from the input of the device (usually the inverting input) to AC ground set the frequency of the pole. In many instances the frequency of this pole is much greater than the expected 3 dB frequency of the closed loop gain and consequently there is negligible effect on stability margin. However, if the feedback pole is less than approximately 6 times the expected 3 dB frequency, a lead capacitor should be placed from the output to the input of the op amp. The value of the added capacitor should be such that the RC time constant of this capacitor and the resistance it parallels is greater than or equal to the original feedback pole time constant.

TYPICAL APPLICATIONS

PNP=2N2905 NPN=2N2219 unless noted TO-5 heat sinks for Q6-Q7 Figure 30. High Speed Current Booster

SNOSBH6D –APRIL 1998–REVISED MARCH 2013 www.ti.com

where AN=1 if the AN digital input is high AN=0 if the AN digital input is low

Figure 31. 10-Bit Linear DAC with No VOS Adjust

Figure 32. Single Supply Analog Switch with Buffered Output

SIMPLIFIED SCHEMATIC

(1) Available per JM38510/11904 Figure 33. Single Supply Analog Switch with Buffered Output

Figure 34.

SNOSBH6D –APRIL 1998–REVISED MARCH 2013 www.ti.com

REVISION HISTORY

Changes from Revision C (March 2013) to Revision D Page

• Changed layout of National Data Sheet to TI format ...... 11

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PACKAGING INFORMATION

Orderable Device Status Package Type Package Pins Package Eco Plan Lead/Ball Finish MSL Peak Temp Op Temp (°C) Device Marking Samples (1) Drawing Qty (2) (6) (3) (4/5) LF411ACN NRND PDIP P 8 40 TBD Call TI Call TI 0 to 70 LF 411ACN LF411ACN/NOPB ACTIVE PDIP P 8 40 Green (RoHS Call TI Level-1-NA-UNLIM 0 to 70 LF & no Sb/Br) 411ACN LF411CN NRND PDIP P 8 40 TBD Call TI Call TI 0 to 70 LF 411CN LF411CN/NOPB ACTIVE PDIP P 8 40 Green (RoHS CU SN Level-1-NA-UNLIM 0 to 70 LF & no Sb/Br) 411CN

(1) The marketing status values are defined as follows: ACTIVE: Product device recommended for new designs. LIFEBUY: TI has announced that the device will be discontinued, and a lifetime-buy period is in effect. NRND: Not recommended for new designs. Device is in production to support existing customers, but TI does not recommend using this part in a new design. PREVIEW: Device has been announced but is not in production. Samples may or may not be available. OBSOLETE: TI has discontinued the production of the device.

(2) Eco Plan - The planned eco-friendly classification: Pb-Free (RoHS), Pb-Free (RoHS Exempt), or Green (RoHS & no Sb/Br) - please check http://www.ti.com/productcontent for the latest availability information and additional product content details. TBD: The Pb-Free/Green conversion plan has not been defined. Pb-Free (RoHS): TI's terms "Lead-Free" or "Pb-Free" mean semiconductor products that are compatible with the current RoHS requirements for all 6 substances, including the requirement that lead not exceed 0.1% by weight in homogeneous materials. Where designed to be soldered at high temperatures, TI Pb-Free products are suitable for use in specified lead-free processes. Pb-Free (RoHS Exempt): This component has a RoHS exemption for either 1) lead-based flip-chip solder bumps used between the die and package, or 2) lead-based die adhesive used between the die and leadframe. The component is otherwise considered Pb-Free (RoHS compatible) as defined above. Green (RoHS & no Sb/Br): TI defines "Green" to mean Pb-Free (RoHS compatible), and free of Bromine (Br) and Antimony (Sb) based flame retardants (Br or Sb do not exceed 0.1% by weight in homogeneous material)

(3) MSL, Peak Temp. - The Moisture Sensitivity Level rating according to the JEDEC industry standard classifications, and peak solder temperature.

(4) There may be additional marking, which relates to the logo, the lot trace code information, or the environmental category on the device.

(5) Multiple Device Markings will be inside parentheses. Only one Device Marking contained in parentheses and separated by a "~" will appear on a device. If a line is indented then it is a continuation of the previous line and the two combined represent the entire Device Marking for that device.

(6) Lead/Ball Finish - Orderable Devices may have multiple material finish options. Finish options are separated by a vertical ruled line. Lead/Ball Finish values may wrap to two lines if the finish value exceeds the maximum column width.

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Appendix C

The homoclinic solution of the hamiltonian part of the slow ﬂow

The slow ﬂow of the system, i.e. a system of two, ﬁrst order, differential equations governed by slow time is given by

λ b 3C J B B¯ a0 = − a + − (a2 + b2)b − + sin( t), 2 2 8 2 2 ω20 λ a 3C A B B¯ b0 = − b − + (a2 + b2)a − − cos( t). (C.1) 2 2 8 2 2 ω20 ¯ where a, b are the variables and A, B, J, , B, ω20 are parameters. The unperturbed part of the above system is b 3C J B a0 = − (a2 + b2)b − + sin γ, 2 8 2 2 a 3C A B b0 = − + (a2 + b2)a − − cos γ (C.2) 2 8 2 2 and γ is a parameter. The equilibrium points are found considering a0 = 0, b0 = 0. After and some simple algebraic manipulations we have A + B cos γ a = b, (C.3) −J + B sin γ and the third order equation

3C (A + B cos γ)2 + (J − B sin γ)2 b − b3 − J + B sin γ = 0. (C.4) 4 (J − B sin γ)2 When (C.4) has three real roots we have three equilibrium points. It is well known that, in order (C.4) to have three real roots it must hold that the determinant q2 + 4p3 < 0, where p = 3ac−b2 2b3−9abc+27a2d 3C (A+B cos γ)2+(J−B sin γ)2 9a2 , q = 27a3 , a = − 4 (J−B sin γ)2 , b = 0, c = 1 and d = −J + B sin γ. This holds for 16 D < (C.5) 81C

145 where D = (A + B cos γ)2 + (J − B sin γ)2. The equilibrium points are 4(A + B cos γ) $ 4(J − B sin γ) $ a1 = − √ cos( ), b1 = √ cos( ), 3 CD 3 3 CD 3 4(A + B cos γ) $ 2π 4(J − B sin γ) $ 2π a2 = − √ cos( + ), b2 = √ cos( + ), 3 CD 3 3 3 CD 3 3 4(A + B cos γ) $ 4π 4(J − B sin γ) $ 4π a3 = − √ cos( + ), b3 = √ cos( + ), (C.6) 3 CD 3 3 3 CD 3 3 √ 1 −1 −9 CD where $ = 3 cos ( 4 ). The hamiltonian of system (C.2) is given by a2 + b2 3C b(B sin γ − J) a h = − (a2 + b2)2 + + (A + B cos γ). (C.7) 4 32 2 2 √ √ We perform the canonical transformation a = 2ρ cos ϑ, b = 2ρ sin ϑ, and the hamiltonian becomes √ ρ 3C 2ρ h = − ρ2 + (sin ϑ(B sin γ − J) + cos ϑ(A + B cos γ)) (C.8) 2 8 2 and √ 2ρ ρ0 = ((B sin γ − J) cos ϑ − sin ϑ(A + B cos γ)). (C.9) 2 √ 2ρ From the square of (C.9) by adding in both sides of the equation the quantity ( 2 ((B sin γ − J) sin ϑ + cos ϑ(A + B cos γ)))2 we have √ 2ρ ρ (ρ0)2 + ( ((B sin γ − J) sin ϑ + cos ϑ(A + B cos γ)))2 = D. (C.10) 2 2 From the hamiltonian (C.8) equation (C.10) becomes √ √ ρD ρ 3C ρD ρ 3C (ρ0)2 = { √ − (h − + ρ2)}{ √ + (h − + ρ2)}. (C.11) 2 2 8 2 2 8 We denote by ρ∗, ϑ∗ the unstable equilibrium point,((ρ∗)0 = 0), and equation (C.10) becomes √ ((B sin γ − J) sin ϑ∗ + cos ϑ∗(A + B cos γ)) = ± D. (C.12) √ By substituting(for the case − D) in the hamiltonian we have r ρ∗ 3C ρ∗D h = − (ρ∗)2 − (C.13) 2 8 2 and (C.11) becomes √ r ρD ρ∗ 3C ρ∗D ρ 3C (ρ0)2 = { √ − ( + (ρ∗)2 − − + ρ2)} 2 2 8 2 2 8 √ r ρD ρ∗ 3C ρ∗D ρ 3C { √ + ( + (ρ∗)2 − − + ρ2)} (C.14) 2 2 8 2 2 8

146 For the right-hand-side of the previous equation after some simple algebra manipulations we have √ r ρD ρ∗ 3C ρ∗D ρ 3C { √ − ( + (ρ∗)2 − − + ρ2)} 2 2 8 2 2 8 √ r ρD ρ∗ 3C ρ∗D ρ 3C { √ + ( + (ρ∗)2 − − + ρ2)} = 2 2 8 2 2 8 r r D D √ 3C √ 3C √ 3C √ (ρ − ρ∗){ ( − ρ∗ + ( ρ∗)3 + ρ∗ρ − ρ∗ρ∗) 2 2 2 4 4 ρ − ρ∗ 3C 9C2 − + (ρ − ρ∗)(ρ + ρ∗) − (ρ − ρ∗)(ρ + ρ∗)2}. (C.15) 4 8 64 We calculate ba0 − ab0 and derive 3C rρ − 2ϑ0ρ = ρ − ρ2 + {sin ϑ(B sin γ − J) + cos ϑ(A + B cos γ)}, (C.16) 2 2 and for the equilibrium point ((ϑ∗)0 = 0) we have

3C rρ∗ ρ∗ − ρ∗2 + {sin ϑ∗(B sin γ − J) + cos ϑ∗(A + B cos γ)} = 0. (C.17) 2 2 Using (C.12) and the above equality (C.15) becomes √ r ρD ρ∗ 3C ρ∗D ρ 3C { √ − ( + (ρ∗)2 − − + ρ2)} 2 2 8 2 2 8 √ r ρD ρ∗ 3C ρ∗D ρ 3C { √ + ( + (ρ∗)2 − − + ρ2)} = 2 2 8 2 2 8 r D 3C √ 1 3C 9C2 (ρ − ρ∗)2{ ρ∗ − + (ρ + ρ∗) − (ρ + ρ∗)2}. (C.18) 2 4 4 8 64 Then from (C.14) and (C.18) we have dρ = dt, (C.19) rq √ ∗ D 3C ∗ 1 3C ∗ 9C2 ∗ 2 |(ρ − ρ )| 2 4 ρ − 4 + 8 (ρ + ρ ) − 64 (ρ + ρ ) that is our main differential equation and is easily solved [17]. As it is seen in Figure (C.1) when our system has three equilibrium points, then depending in the parameters, we may have two homoclinic orbits. In our analysis this result is given by the absolute value in (C.19). For the case ρ∗ > ρ the homoclinic solution is

Q(t) 3√ ∗ Q(t) ∗ ∗2 √ (128e 2 + 27C 2Dρ∗)ρ − 16e 4 (3C(6ρ − 3Cρ + 4 2Dρ∗) − 8) ρ(t) = Q(t) √ Q(t) (C.20) 128e 2 + 27C3 2Dρ∗ − 48e 4 (3Cρ∗ − 2) p √ where Q(t) = t −4 + 3C(2 2Dρ∗ + ρ∗(4 − 3Cρ∗)). After substituting the solution (C.20) in (C.16) and integrate we derive

−1 Q(t) −1 Q(t) ϑ = W1 tan (g1 + g2e 4 ) + W2 tan (g3 + g4e 4 ) + W3t, (C.21)

147 1.0

0.5

0.0

-0.5

-1.0

-1.0 -0.5 0.0 0.5 1.0 π Figure C.1: phase space of the hamiltonian for A = 0.1,B = 0.001,J = 0.00001, γ = 9 and C = 2.0. where √ 3C(8 − 6(1 + 3C)ρ∗ + 9C(1 + C)ρ∗ − 12C 2Dρ∗) W1 = q √ q √ , 2 ∗ 2 2 3 ∗ 2 2 2 2 ∗ −9C (ρ − 3C ) + 6C 2Dρ −9C (ρ − 3C ) + 6C 2Dρ h W2 = q √ ∗ 2 ∗ 2 2 3 ∗ ρ −9C (ρ − 3C ) + 6C 2Dρ √ 6(3C − 1)ρ∗ − 9C(C − 1)ρ∗ + 12C 2Dρ∗ − 8 √ √ , p54C3ρ∗2 2Dρ∗ − (8 + 3C(3ρ∗(Cρ∗ − 2) − 4 2Dρ∗))2 9Cρ∗2 − 16ρ∗ − 8h W = , 3 64ρ∗ 2 − 3Cρ∗ 16 g1 = q √ , g2 = q √ , 2 ∗ 2 2 3 ∗ 2 ∗ 2 2 3 ∗ −9C (ρ − 3C ) + 6C 2Dρ 3 −9C (ρ − 3C ) + 6C 2Dρ √ 8 + 9C2ρ∗2 − 6C(3ρ∗ + 2 2Dρ∗) g3 = √ √ , p54C3ρ∗2 2Dρ∗ − (8 + 3C(3ρ∗(Cρ∗ − 2) − 4 2Dρ∗))2 16ρ∗ g4 = √ √ . (C.22) p54C3ρ∗2 2Dρ∗ − (8 + 3C(3ρ∗(Cρ∗ − 2) − 4 2Dρ∗))2

For the case ρ > ρ∗ the homoclinic solution is

− Q(t) 3√ ∗ −Q(t) ∗ ∗2 √ (128e 2 + 27C 2Dρ∗)ρ − 16e 4 (3C(6ρ − 3Cρ + 4 2Dρ∗) − 8) ρ(t) = −Q(t) √ −Q(t) ,(C.23) 128e 2 + 27C3 2Dρ∗ − 48e 4 (3Cρ∗ − 2) and −1 − Q(t) −1 − Q(t) ϑ = W3t − W1 tan (g1 + g2e 4 ) − W2 tan (g3 + g4e 4 ). (C.24)

148 Bibliography

[1] Arrowsmith D.K., Place C.M., "An Introduction to Dynamical Systems", Cambridge Uni- versity Press, 1994.

[2] Bountis A., "Dynamical systems and chaos, volume A ", Papasotiriou, 1995, (In Greek).

[3] Bountis A., "Dynamical systems and chaos, volume B", University of Patras, 2000, (In Greek).

[4] Bountis T., Skokos H., "Complex Hamiltonian Dynamics", Springer, 2012.

[5] Carroll, T., Pecora, L., (Editors), ”Nonlinear Dynamics in Circuitsb“, World Scientiﬁc, 1995.

[6] G. Chen, and X. Dong,“From chaos to order: methodologies, perspectives, and applications”, Singapore: World Scientiﬁc, 1998.

[7] G. Chen, and X. Yu,“Chaos control: theory and applications”, Berlin, Heidelberg: Springer- Verlag, 2003.

[8] L.O. Chua, Archiv fur Elektronik und Ubertragungstechnik vol.46, 250, 1992.

[9] Chua, L. O., Desoer, C. A., Kuh, E. S., “Linear and Nonlinear Circuits”, McGraw-Hill, 1987.

[10] Davis H.T., “Introduction to nonlinear differential and integral equations”, Dover, 1962.

[11] A.S. Dimitriev, A.V. Kletsovi, A.M. Laktushkin, A.I. Panas, and S.O. Starkov, “Ultraw- ideband wireless communications based on dynamic chaos”, J. Communications Technology Electronics vol.51, 1126 (2006).

[12] Gendelman O.V, Manevitch L.I, Vakakis A.F, M’Closkey R., “Energy Pumping in Nonlinear Mechanical Oscillators: Part I- Dynamics of the Underlying Hamiltonian Systems”, J. Appl. Mech. vol. 68, 34-41, 2001.

[13] Gendelman, O.V., “Bifurcations of nonlinear normal modes of linear oscillator with strongly nonlinear damped attachment”, Nonlinear Dynamics vol.37, 115-128, 2004.

149 [14] Gendelman O.V., “Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators”, Nonlinear Dynamics vol.25, 237-253, 2001.

[15] Gendelman O. V., Starosvetsky Y., Feldman M., Attractors of harmonically forced linear oscillator with attached nonlinear energy sink I: Description of response regimes, Nonlinear Dyn, vol.51, 31-46 (2008).

[16] Gendelman O.V., Vakakis A.F., Bergman L.A., McFarland D.M., Asymptotic Analysis of Passive Suppression Mechanisms for Aeroelastic Instabilities in a Rigid Wing in Subsonic Flow, SIAM Journal on Applied Mathematics, vol. 70, No. 5, 1655-1677 (2010).

[17] Gradshteyn I.S., Ryzhik I.M., Table of Integrals, Series, and Products, Seventh Edition, Academic Press, 2007.

[18] Guckenheimer J., Hoffman K., Weckesser W., “Bifurcations of relaxation oscillations near folded saddles”, Int. J. Bif. Chaos, vol. 15, 3411-3421, 2005.

[19] Guckenheimer J., Wechselberger M., Young L.S., “Chaotic attractors of relaxation oscillators”, Nonlinearity, vol. 19, 701-720, 2006.

[20] Guckenheimer J. and Holmes P., “Nonlinear Oscillations, Dynamical Systems, and Bifur- cations of Vector Fields”, Applied Mathematical Sciences vol. 42, Springer,1983.

[21] Fenichel N., Geometric singular perturbations theory for ordinary differential equations, J. Differ. Equ, 31, 53-98 (1979).

[22] Hadjidemetriou J., "Periodic orbits and stability", Erasmus Program, 1988.

[23] Hadjidemetriou J., "Theoretical Mechanics", Giaxoudi - Giapouli, 1983, (In Greek).

[24] Hadjidemetriou J., "Stability of dynamical systems", proceedings of the 11th summer school of nonlinear dynamics, Livadia, 2000, (In Greek).

[25] Hadjidemetriou J., "Structural stability", Order and chaos in nonlinear dynamical systems, G.S. Pnevmatikos, Athens 2002, (In Greek).

[26] Hadjidemetriou J., “The Poincare Map and the Method of Averaging: A Comparative Study”, NONLINEAR PHENOMENA IN COMPLEX SYSTEMS, An Interdisciplinary Journal, vol. 11, 149-157, 2008.

[27] Holmes M., “Introduction to Perturbation Methods, second edition”, Springer Verlag, 2013.

[28] Hunter J.K., “Asymptotic Analysis and Singular Perturbation Theory”, Department of Mathematics University of California at Davis, 2004.

[29] Ichtiaroglou S., "Integrability and non- integrability in Planar Hamiltonian Systems", Erasmus program, 1989.

150 [30] Ichtiaroglou S.,"Introduction to Hamiltonian Mechanics", Aristotle University of Thessa- loniki, 2004, (In Greek).

[31] Ichtiaroglou S., Hadjidemetriou J., “ Dynamical systems and chaos”, Aristotle University of Thessaloniki, 2000, (In Greek).

[32] T. Jiang, S. Qiao, Z. Shi, L. Peng, and J. Huangfu, “Simulation and experimental evaluation of the radar signal performance of chaotic signals generated from a microwave Colpitts oscillator”, Progress In Electromagnetics Research vol.90, 15 2009.

[33] Jones C.K.R.T., “Geometric Singular Perturbation Theory”, Dynamical Systems, Lecture Notes in Mathematics, Springer Verlag,1995.

[34] Kapitaniak, T. (Editor), “Chaotic Oscillators. Theory and Applications”, World Scientiﬁc, 1992.

[35] Karagiannis I, Theodossiades S, “Targeted energy transfer in hypoid gears of automotive differentials”, ENOC 2011.

[36] Kerschen G., Vakakis A.F.,Lee Y.S., McFarland D.M., Kowtko J.J., Bergman L.A., “Energy transfers in a system of two coupled oscillators with essential nonlinearity: 1:1 resonance manifolds and transient bridging orbits”, Nonlinear Dynamics, vol.42, 283-303,2005.

[37] Kevorkian J., Cole J.D., “Multiple scale and singular perturbation methods”, Springer Verlag, 1996.

[38] Kyprianidis I.M, Petrani M.L., “Nonlinear electric circuits”, Sychrony Paideia, 2000 (In Greek).

[39] I. M. Kyprianidis, Ch. K. Volos, I. N. Stouboulos, and J. Hadjidemetriou, “Dynamics of two Resistively Coupled Dufﬁng-Type Electrical Oscillators ”, Int. J. Bifurc. Chaos vol.16, 1765, 2006.

[40] Klinker, T., Meyer-Ilse, W., Lauterborn, W., “Period Doubling and Chaotic Behavior in a Driven Toda Oscillator”, Phys. Lett. A, vol.101 (1984) 371.

[41] Lacy, J. G., “A Simple Piecewise-Linear Non-Autonomous Circuit with Chaotic Behavior“, Int. J. Bifurc. Chaos, vol.6 (1996) 2097.

[42] Maaita J.O., Meletlidou E., Vakakis A.F., Rothos V., ”The effect of Slow Flow Dy- namics on the Oscillations of a singular damped system with an essentially nonlinear attachment“, Journal of Applied Nonlinear Dynamics, vol. 2, issue 4, 315-328, DOI: 10.5890/JAND.2013.11.001, 2013.

151 [43] Maaita J.O., Meletlidou E., Vakakis A.F., Rothos V., “The dynamics of the slow ﬂow of a singular damped nonlinear system and its Parametric Study”, Journal of Applied Nonlinear Dynamics, vol. 3, issue 1, 37 - 49, DOI: 10.5890/JAND.2014.03.004 .

[44] Maaita J.O. and Meletlidou E, “Analytical Homoclinic Solution of a Two-Dimensional Nonlinear System of Differential Equations“, Journal of Nonlinear Dynamics, vol. 2013, Article ID 879040, 4 pages, 2013. doi:10.1155/2013/879040

[45] Maaita J.O., Kyprianidis I.M., Volos Ch. K, Meletlidou E., ”The Study of a Nonlinear Dufﬁng - Type Oscillator Driven by Two Voltage Sources“, Journal Of Engineering Science And Technology Review, vol.6, issue 4, p. 74-80, 2013.

[46] L. I. Manevitch, Complex representation of dynamics of coupled oscillators, in Mathemati- cal Models of Nonlinear Excitations, Transfer Dynamics and Control in Condensed Systems, 269-300, Kluwer Academic Publishers/Plenum, New York (1999).

[47] I. Manimehan, K. Thamilmaran, and P. Philominathan, “torus Breakdown To Chaos Via Period-3 Doubling Route In A Modiﬁed Canonical Chua’S Circuit ” Int. J. Bifurc. Chaos vol.21, 1987 2011.

[48] McFarland D.M., Bergman L.A, Vakakis A.F., “Experimental study of non-linear energy pumping occurring at a single fast frequency”, Int. J. Nonlinear Mech. vol.40, 891-899, 2005.

[49] Nayfeh A., “Perturbation Methods“, Wiley-Interscience publication, 1973.

[50] Neishtadt A.I., On the change in the adiabatic invariant on crossing a separatix in systems with two degrees of freedom, PMM USSR, 51, No.5, 586-592 (1987).

[51] Nucera F., Lo Iacono F., McFarland D.M., L.A. Bergman L.A., Vakakis A.F., “Application of broadband nonlinear targeted energy transfers for seismic mitigation of a shear frame: Experimental results”, J. of Sound and Vibration vol.313, 57-76,2008.

[52] Ogorzalek, M. J., “Chaos and Complexity in Nonlinear Electronic Circuits”, World Scien- tiﬁc, 1997.

[53] Panagopoulos P.N., Gendelman O.V., Vakakis A.F., “Robustness of nonlinear targeted energy transfer in coupled oscillators to changes of initial conditions”, Nonlinear Dynamics, vol. 47, 377-387,2007.

[54] D. Quinn, O. Gendelman, G. Kerschen, T. Sapsis, L. Bergman, A. Vakakis., “Efﬁciency of Targeted Energy Transfers in Coupled Nonlinear Oscillators Associated with 1:1 Resonance Captures: Part I”, J. of Sound and Vibration, vol.311 (2008) 1228.

[55] T. Sapsis, A. Vakakis, O. Gendelman, L. Bergman, G. Kerschen, D. Quinn., “Efﬁciency of targeted energy transfers in coupled nonlinear oscillators associated with 1:1 resonance captures: Part II”, Analytical study, J. of Sound and Vibration, vol.325 (2009) 297-320.

152 [56] Skokos Ch., “The Lyapunov Characteristic Exponents and their computation”, Lecture Notes in Physics, vol. 790, 63-135, Springer- Verlag, 2010.

[57] S.H. Strogatz,“Nonlinear Dynamics and Chaos”, Colorado: Westview Press, 1994.

[58] C.K. Tse, and F. Lau,“Chaos-based Digital Communication Systems”, Berlin, New York: Springer Verlag, 2003.

[59] Tikhonov A.N., Systems of differential equations containing a small parameter multiplying the derivative, Mat. Sb., 31, 575-586 (1952).

[60] Tripepi C., Nucera F., Vakakis A.F., Bergman L., McFarland D.M., “Seismic protection of seismically excited symmetric and eccentric steel structures through targeted energy transfer, EUROMECH Colloquium 503, 2009.

[61] Udwadia F., E., Von Bremen H. F., “Computation of Lyapunov characteristic exponents for continuous dynamical systems”, Zeitschrift fur angewandte Mathematik und Physik ZAMP, vol. 53, Springer- Verlag, 2002.

[62] Ueda, Y. and Akamatsu, N., “Chaotically Transitional Phenomena in the Forced Negative Resistance Oscillator”, IEEE Trans. Circuits Syst., vol. CAS-28 (1981) 217.

[63] Vakakis A.F., Gendelman O.V., Bergman L.A., McFarland D.M., Kerschen G., Lee Y.S., “Nonlinear Target Energy Transfer in Mechanical and Structural Systems”, Springer Verlag, 2008.

[64] Vakakis A.F., “Inducing Passive Nonlinear Energy Sinks in Vibrating Systems”, J. Vib. Acoust. vol.123, 324-332, 2001.

[65] Vakakis A.F, Gendelman O.V, “Energy Pumping in Nonlinear Mechanical Oscillators: Part II- Resonance capture”, J. Appl. Mech, vol.68, 34-41, 2001.

[66] Vakakis A.F., “Relaxation oscillations, subharmonic orbits and chaos in the dynamics of a linear lattice with a local essentially nonlinear attachment”, Nonlinear Dynamics,vol. 61, 443-463, 2010.

[67] Van Buskirk, R., Jeffries, C., “Observation of Chaotic Dynamics of Coupled Nonlinear Oscillators”, Phys. Rev. A, vol.31 (1985) 3332.

[68] Vakakis A.F., Relaxation Oscillations, Subharmonic Orbits and Chaos in the Dynamics of a Linear Lattice with a Local Essentially Nonlinear Attachment, Nonlinear Dynamics, 61, 443-463 (2010).

[69] Verhulst F, “Singular perturbation methods for slow - “fast dynamics”, Nonlinear Dynamics, vol.50, 747-753, 2007.

153 [70] Voyatzis G., Meletlidou E., "Introduction to computational of dynamical systems" Aristotle University of Thessaloniki, 2005, (In Greek).

[71] Viguie R., Kerschen G, Golinval J.C., McFarland D.M., Bergman L.A., Vakakis A.F., van de Wouw N., “Using Targeted Energy Transfer to Stabilize Drill-string Systems”, Mechanical Systems and Signal Processing, vol.23, 148-169,2009.

[72] Ch.K. Volos, I.M. Kyprianidis, and I.N. Stouboulos, “ Experimental demonstration of a chaotic cryptographic scheme”, WSEAS Trans. Circuits Syst. vol.5, 1654 (2006).

[73] Ch.K. Volos, I.M. Kyprianidis, and I.N. Stouboulos, “Image encryption process based on chaotic synchronization phenomena”, Signal Processing vol.93, issue 5, 1328, 2013.

[74] Ch.K. Volos, I.M. Kyprianidis, and I.N. Stouboulos, “Fingerprint images encryption process based on a chaotic true random bits generator”, Int. J. Multimedia Intelligence and Security vol.1, 320 2010.

[75] Ch.K. Volos, I.M. Kyprianidis, and I.N. Stouboulos, “Motion Control of Robots Using a Chaotic Truly Random Bits Generator ”, Journal of Engineering Science and Technology Review vol.5, issue 2, 6, 2012.

[76] Ch. K. Volos, I. M. Kyprianidis, I. N. Stouboulos, and A. N. Anagnostopoulos, “experimental Study Of The Dynamic Behavior Of A Double Scroll Circuit”, J. Applied Functional Analysis vol.4, 703, 2009.

[77] Wiggins S., “Global Bifurcations and Chaos, Analytical Methods”, Applied Mathematical Sciences, Springer Verlag, 1988.

[78] A. Wolf., J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series”, Physica D 16, 285 (1985).

154