©2020 Chaitanya Borra All Rights Reserved

CHAITANYA BORRA

A Dissertation

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy

Chaitanya Borra

May, 2020 DYNAMICS OF LARGE ARRAY MICRO/NANO RESONATORS

Chaitanya Borra

Dissertation

Approved: Accepted:

Advisor Department Chair D. Dane Quinn Sergio Felicelli

Committee Member Dean of the College Graham Kelly Craig Menzemer

Committee Member Dean of the Graduate School Ernian Pan Marnie Saunders

Committee Member Date Patrick Wilber

Committee Member Kwek Tze Tan

ii ABSTRACT

This work describes an analytical framework suitable for the analysis of large-scale arrays of coupled resonators, including those which feature amplitude and phase dynamics, inherent element-level parameter variation, nonlinearity, and/or noise. In particular, this analysis allows for the consideration of coupled systems in which the number of individual resonators is large, extending as far as the continuum limit corresponding to an infinite number of resonators. Moreover, this framework per- mits analytical predictions for the amplitude and phase dynamics of such systems.

The utility of this analytical methodology is explored through the analysis of a system of N non-identical resonators with global coupling, including both reactive and dissipative components, physically motivated by an electromagnetically-transduced microresonator array. In addition to the amplitude and phase dynamics, the behavior of the system as the number of resonators varies is investigated and the convergence of the discrete system to the infinite-N limit is characterized.

The equilibrium obtained by the array of resonators may be unique if the behavior of many identical or slightly different resonators in the population is small, such equilibrium is also called mono-stable equilibrium. The dynamics of such systems can be identified with a typical single degree of freedom systems under the

iii influence of harmonic excitations at different frequencies. The behavior ofanarray of resonators can be described using two self-consistency variables and a force, which not only includes the external excitation but also due to the reactive and dissipative coupling parameters, this is called a global coupling function. The behavior of self-consistency variables and global coupling forces in the case of a linear system is largely monotonous for the variations in the coupling parameters.

Unlike monostable and linear systems, the effect of reactive and dissipative coupling on non-linear systems cannot be decoupled and is not so straightforward.

Duffing like behavior can be observed by increasing the external excitations ornon- linear stiffness. In order to study the effect of reactive coupling, on these systems, the dissipative coupling is held constant and reactive coupling is varied, it is observed that multiple solutions start appearing when the reactive coupling is increased. This is mainly due to the resonators with their detuning close to the resonant actuating frequency, this localized phenomenon is similar to a Duffing response. Two different bifurcations resulting from the variation of coupling parameters have been identified, which explains the underlying phenomenon that leads to multiple-equilibrium distributions. The stability of distributions is found by evaluating the eigenvalues, from the discrete spectrum equation.

iv DEDICATION

I would like to dedicate this work to my Grandfather(Surya Prakasa Rao), my Par- ents(Rama Mohan and Manikya Kumari), my wife(Lakshmi Kalyani) and my younger brother(Vamsi).

v ACKNOWLEDGEMENTS

I would like to take this opportunity to thank my advisor Dr. Dane. D. Quinn for his guidance during my research. I am grateful to Dr. Graham Kelly, Dr. Kwek Tze

Tan, Dr. Ernian Pan, and Dr. Patrick Wilber for serving on my committee.

I am very thankful to National Science Foundation for supporting my dissertation research (Grant number 1537701). I am also thankful to Dr. Quinn and

Dr.Sergio Felicelli for providing the funding and assistantship opportunity.

A special thanks to my fellow students Allen, Ali, Dushyanth, Sunny and others whose friendships have made my experience at the University of Akron more enjoyable. I would also like to express my appreciation to the current and former staff of the Mechanical Engineering and Aerospace department including Shannon, Ellen,

Stacy and Lisa, for their help and friendship during my tenure at the University of

Akron.

I would like to thank my family and friends for their assistance during my academic career. Finally, and most importantly, I would like to express my deepest gratitude to my parents, wife and my brother whose love, support and encouragement made this dissertation possible.

vi TABLE OF CONTENTS

Page

LIST OF FIGURES ...... ix

CHAPTER

I. INTRODUCTION ...... 1

1.1 Single Resonator Dynamics ...... 4

1.2 Coupled resonators ...... 11

1.3 Organization of the Thesis ...... 19

II. LITERATURE ...... 22

2.1 Coupled Resonators ...... 24

2.2 Mathematical models ...... 29

III. ANALYTICAL MODELING OF LARGE ARRAY RESONATORS . 39

3.1 A Discrete System of Coupled Resonators ...... 39

3.2 Continuum Model ...... 41

3.3 Results ...... 49

IV. MONOSTABLE DISTRIBUTIONS ...... 56

4.1 Introduction ...... 56

4.2 Results ...... 57

4.3 Conclusions ...... 67

vii V. EFFECT OF COUPLING PARAMETERS ON MULTIPLE EQUILIBRIUM STATES ...... 68 5.1 Monostable-Unique Solutions ...... 69

5.2 Multiple Equilibrium Distributions ...... 71

5.3 Non-isolated Equilibrium states ...... 101

5.4 Stability region ...... 125

5.5 Summary ...... 130

VI. CONCLUSION AND FUTURE WORK ...... 134

6.1 Summary ...... 134

6.2 Suggested Future work ...... 139

BIBLIOGRAPHY ...... 140

viii LIST OF FIGURES

Figure Page

1.1 Schematic diagram of beam element used as resonator, here u, v, and ψ are longitudinal, transverse and angular displacements of the differential beam element [1]...... 5

1.2 F0 = 2.0 , ζ = 0.1 , γ = 0.1 a)Amplitude R(σ) vs σ b) Phase ϕ(σ) vs σ 9

1.3 F0 = 2.0 , ζ = 0.1 a)Amplitude R(σ) vs σ b) Phase ϕ(σ) vs σ, γ = −0.5, γ = −0.1, γ = 0, γ = 0.1, γ = 0.5. . . . . 10

1.4 γ = 0.1 , ζ = 0.1 Frequency response R(σ) vs σ at F0 = 0.5, F0 = 1.0, F0 = 3.0, backbone curve representing the 3γR2 locus of peak amplitudes(σ = 8ω )...... 11

1.5 γ = 0.1 ,F0 = 2.0 Frequency response R(σ) vs σ at ζ = 0.05, ζ = 0.1, ζ = 0.2, ζ = 0.5...... 12 1.6 Coupled system of resonators with 2 degree of freedom systems . . . . . 12

1.7 F0 = 2.0 , ζ1 = ζ2 = 0.25 , αg = 0 R(1)( σ1 = −5.0), R(2)( σ2 = 5.0) vs Ω (excitation frequency) for (a) βg = 0 , (b) βg = 0.5 , (c) βg = 5.0 and (d) βg = 10.0 ...... 16

1.8 F0 = 2.0 , ζ1 = ζ2 = 0.25 , βg = 0.5 R(1)( σ1 = −5.0), R(2)( σ2 = 5.0) vs Ω (excitation frequency) for (a) αg = 0 , (b) αg = 0.5 , (c) αg = 2.0 and (d) αg = 10.0 ...... 17

1.9 F0 = 2.0 , ζ1 = ζ2 = 0.25 , βg = 2.0 , αg = 0.2 R(1)( b b σ1 = −5.0), R(2)( b b σ2 = 5.0) vs Ω (excitation frequency) for (a) γ = 0 , (b) γ = 0.08 , (c) γ = 0.5 and (d) γ = 2.0 ...... 18

2.1 SEM image of two resonant filter [2], series-coupled resonator deigned for operation at atmospheric pressure...... 25 2.2 SEM image of polysilicon fabricated micromechanical disk-arrays [3],a) Top-view b) Iso-view image...... 26

ix 2.3 SEM image of four 500µm × 4µm clamped-clamped array beams with 9µm spacing [4], through-support coupling was extended to large arrays simply placing beams adjacent to each other...... 27 2.4 Scanning electron micrograph of a representative electromagnetically transduced microresonator. The device consists of an electrically and mechanically isolated silicon microcantilever and two Au/Cr wire loops, which follow the perimeter of the microcantilever [5]...... 30

3.1 Response for a mono-stable solution γ = 0.02, bc bc , for coupling parameters (α , β) = (0.5 , 1.0)...... 50

3.2 R(s) vs m(s) for global coupling constants (V,W ) = (0.3840 , −0.3298), given parameters γ = 0.2 , (α , β) = (0.5 , 1.0) X(s) > 0, X(s) > 0(unstable branch), X(s) = 0, X(s) < 0 ...... 51

bc bc E bc bc E U bc bc E 3.3 Multiple solutions at γ = 0.08, 6, 6 , 7, for coupling parameters (α , β) = (0.5 , 1.0) ...... 53

3.4 Re(λ) vs Im(λ), γ = 0.02 (α , β) = (0.5 , 1.0) bc bc Discrete spectrum. 54

E E U E 3.5 Spectrum plot Re(λ) vs Im(λ), γ = 0.08 a) 6, b) 6 and c) 7 , E E E U both 6 and 7 have eigenvalues with Re(λ) < 0; 6 has both the spectrums with Re(λ) > 0 ...... 55

4.1 Linear response functions (α = 2.00, β = 5.00, γ = 0, N = 100); iterative solution, ⊗ ⊗ ⊗ direct numerical simulation; (a) R(s), (b) Φ(s)...... 59

4.2 Coupling kernels (α = 2.00, β = 5.00, γ = 0, N = 100, iterative solution); (a) A(s) ρ(s), (b) B(s) ρ(s)...... 59

4.3 Linear response functions varying α (β = 10.00, γ = 0, N = 100); α = 0 , α = 4.0 , α = 8.0; (a)R(s), (b)Φ(s) ...... 60

4.4 Linear response functions varying β (α = 4.00, γ = 0, N = 100); β = 0 , , β = 10.0 , β = 20.0; (a)R(s), (b)Φ(s) ...... 61

4.5 Maximum response amplitude Rmax, varying coupling parameters (γ = 0, N = 100); iterative solution, bC bC bC direct numerical simulation; (a) dissipative coupling (β = 0), (b) reactive coupling (α = 0)...... 61

x 4.6 Global coupling force G∗ acting on all the resonators in the array, varying coupling parameters (γ = 0, N = 100); iterative solution, bC bC bC direct numerical simulation; (a) dissipative coupling (β = 0), (b) reactive coupling (α = 0)...... 62

4.7 Nonlinear response functions, varying γ (α = 2.00, β = 5.00, N = 100, iterative solutions); ut ut γ = −0.10, bc bc γ = 0, rs rs γ = 0.10; (a) R(s), (b) Φ(s)...... 63

4.8 Response functions (α = 2.00, β = 5.00, γ = 0.05, iterative solutions); N = 1000, bC bC bC N = 20; (a) R(s), (b) Φ(s)...... 64

4.9 Global self-consistency coupling variables, varying N (α = 2.00, β = 5.00, γ = 0, iterative solutions); (a) V , (b) W ...... 65

4.10 Global self-consistency coupling variables (α = 2.00, β = 5.00, γ = 0, iterative solutions, 500 random distributions); (a) N = 20, (b) N = 500. The large point indicates the values obtained from the uniform distribution of resonators...... 66 4.11 Global self-consistency coupling variables, iterative solutions, normal distribution varying σ (α = 2.00, β = 5.00, γ = 0, N = 500); (a) V , (b) W ...... 66

4.12 A curve separating the stable(left) and unstable(right) regions on an α , β plane, for an uniform distribution, γ = 0, N = 100...... 67

5.1 a) Equilibrium distribution showing states of resonators(R(s) vs m(s)) b) Global self consistency constants (V,W ) = (0.235 , −0.1813), at the coupling values of (α , β) = (−1.0 , −2.0), only an unique solution state exists, bc bc , E7 indicates the state of individual resonators in the equilibrium distribution...... 70

5.2 a) Equilibrium distributions after bifurcation, referred as Type-I saddle-node bifurcation points for coupling parameters (α , β) = (−0.3 , 1.2) b) Enlarged image showing resonator s7 on same bc bc E U bc bc E U branches 7 (unstable), 7 ...... 72 5.3 a) Equilibrium distributions after Type II bifurcation for coupling parameters (α , β) = (0.1 , 5.3) b) Enlarged image of resonator, bc bc E bc bc E U 9, 9 ...... 73 5.4 a) Equilibrium distributions showing states of resonators(R(s) vs bc bc E bc bc E bc bc E U m(s)) 7, 8 and 7 , b) Global self consistency (V,W ), at the coupling values of (α , β) = (−1.0 , 2.0)...... 74

xi − E E U 5.5 Spectrum plot Re(λ) vs Im(λ), (α , β) = ( 1.0 , 2.0) a) 7, b) 7 E E E E U and c) 8 , both 7 and 8 have eigenvalues with Re(λ) < 0; 7 has eigenvalue Re(λ) > 0 ...... 76

5.6 Self-Consistency variables (V,W ) as a function of reactive coupling − E E E E U (β) at α = 1.0, 7, 8, 8(unstable) , 7 , Figure (a) and (b) shows two different orthogonal planes of Figure (c), ′ points P and P indicate Type-I bifurcation ...... 77

bc bc E U 5.7 Two similar distributions of 7 near the point P (α , β) = (−1.0 , 0.06), enlarged picture here shows resonator 7 in two distributions, indicating Type-I bifurcation...... 78

′ 5.8 Two similar distributions of bc bc E8 near the point P (α , β) = (−1.0 , 3.06), enlarged image here shows resonator 7 in two distributions, indicating Type-I bifurcation...... 79

5.9 Response of resonator s7 (R(s7) vs β) for the distributions E7, U ′ E , E8, E8(unstable), Q and Q indicate the branch 7 ′ change points, P and P indicate the points of Type-I bifurcation. . . . 80

U 5.10 X(s7) vs β for equilibrium distributions E7, E , E8, ′ 7 E8 (unstable), points P and P indicate the bifurcation of Type I, E U E Q is the point where the distribution 7 transforms to 8(stable), ′ E U E while Q is the point where the distribution 7 transforms to 8 (unstable)...... 82

E U E 5.11 Response of equilibrium distributions, 7 , 8, they transform with variation in β at α = −1.0...... 84

5.12 Spectrum plot for (α , β) = (−1.0 , 2.5) a) Equilibrium state E8 is stable, b) Newly transformed equilibrium state E8 is unstable...... 85 5.13 Self-Consistency variables (V,W ) as a function of reactive coupling E E E E U (β) at different α, 7, 8, 8(unstable) , 7 , all the bifurcations that are shown here are of Type-I ...... 87

5.14 Global coupling force G∗ as a function of reactive coupling (β) at E E E E U different α, 7, 8, 8(unstable) , 7 , all bifurcations are Type I...... 88

E E U − 5.15 Response for 9 , 8 at (α , β) = ( 0.3 , 1.0) ...... 89

E U E 5.16 Equilibrium distributions represented by, 8 , 9(unstable), E9, α = −0.3 a) Amplitude of resonator R(s8) vs β, b) X(s8) vs β . 90

xii 5.17 Amplitude of resonator s8 R(s8) vs β for equilibrium distributions E U E E E U 8 , 9, 9(unstable), 9 , at different α ...... 91 5.18 Self-Consistency variables (V,W ) as a function of reactive coupling E U E E E U (β) at different α, 8 , 9, 9 (unstable), 9 , both Type-I and Type-II can be seen here...... 93

5.19 Global coupling force G∗ as a function of reactive coupling (β) at E U E E E U different α, 8 , 9, 9 (unstable) 9 , both Type-I and Type-II can be seen here ...... 94

5.20 At α = 0.2, the equilibrium distributions influenced by resonator s8 bc bc E bc bc E U are 9, 9 , a) at β = 6.8 two of the distributions are of E9 type, b) at higher β resonator s9(from stable distribution of E9) changes from stable part of X(s) > 0 to unstable part of continuous E E U Duffing curve in 9 distribution transforms to 9 ...... 95 E E U 5.21 Response for ∞ , 10 at (α , β) = (0.08 , 2.0) ...... 96

E U E E 5.22 Equilibrium distributions represented by, 10, ∞, ∞ (unstable) a) Amplitude of resonator R(s10) vs β, b) X(s10) vs β at α = 0.08, Type I bifurcation ...... 97

5.23 Self-Consistency variables (V,W ) as a function of reactive coupling E U E E (β) at different α, 10, ∞, ∞ (unstable), Type I bifurcation. 98

5.24 Global coupling force G∗ as a function of reactive coupling (β) at E U E E different α, 10, ∞, ∞ (unstable), Type-I bifurcation can be seen here...... 99

5.25 Self-Consistency variables (V,W ) as a function of reactive coupling (β) at α = 0.08, a) 3-D plot of (V,W ) vs β b) 2-D plot of E E U E (V,W ), Open Contour ( 7), Closed Contour-1 ( 7 , 8, E E U E E 8 (unstable)), Closed Contour-2 ( 8 , 9, 9 (un- E U E E stable)), Closed Contour-3 ( 9 , 10, 10 (unstable)) , E U E E Closed Contour-4 ( 10, ∞, ∞ (unstable))...... 100 E U E E 5.26 G∗ as a function of β in Closed Contour-3 ( 9 , 10, 10 (unstable)) at α = 0.08 ...... 102 5.27 Different equilibrium distributions that occur for discrete points 1 5 from C3 to C3 and on the curve shown in Figure 5.26 at α = 0.08, bc bc E U bc bc E 9 , 10...... 104

xiii E U E E 5.28 G∗ as a function of β in Closed Contour-3 ( 9 , 10, 10 E U (unstable), 10) at α = 0.2 ...... 105 5.29 Different equilibrium distributions that occur for discrete points 1 4 from C3 to C3 and on the curve shown in Figure 5.28 at α = 0.2, bc bc E U bc bc E 10, 10 ...... 106 5.30 Different equilibrium distributions that occur for discrete points 5 7 from C3 to C3 and on the curve shown in Figure 5.28 at α = 0.2, bc bc E U bc bc E 9 , 10 ...... 108 5.31 Self-Consistency variables (V,W ) as a function of reactive coupling (β) at α = 0.2, (a) 3-D plot of (V,W ) vs β b) 2-D plot of E E E U (V,W ), Open Contour ( 7, 6, 7 ), Closed Contour- E U E E E U 1 ( 7 , 8, 8 (unstable), 8 ), Closed Contour- E U E E E U 2 ( 8 , 9, 9(unstable), 9 ), Closed Contour-3 E U E E E U ( 9 , 10, 10 (unstable), 10), Closed Contour-4 E U E E ( 10, ∞, ∞ (unstable))...... 109

5.32 Global coupling force G∗ as a function of reactive coupling (β) at E E E U α = 0.2 Open Contour ( 7, 6, 7 ), Closed Contour- E U E E E U 1 ( 7 , 8, 8 (unstable), 8 ), Closed Contour- E U E E E U 2 ( 8 , 9, 9(unstable), 9 ), Closed Contour-3 E U E E E U ( 9 , 10, 10(unstable), 10) , Closed Contour-4 E U E E ( 10, ∞, ∞(unstable))...... 110 E U E E 5.33 G∗ as a function of β in Closed Contour-3 ( 9 , 10, 10 E U (unstable), 10) at α = 0.3 ...... 111 5.34 Different equilibrium distributions that occur for discrete points 1 4 from C3 to C3 and on the curve shown in Figure 5.33 at α = 0.2, bc bc E bc bc E U 10, 10...... 113 5.35 Different equilibrium distributions that occur for discrete points 5 9 from C3 to C3 and on the curve shown in Figure 5.33, at α = 0.2 bc bc E U bc bc E U bc bc E 10, 9 , 10...... 114 E U E 5.36 G∗ as a function of β in Closed Contour-4 ( 10, ∞, E∞(unstable)) at α = 0.3 ...... 116 5.37 Different equilibrium distributions that occur for discrete points 1 5 from C4 to C4 and on the curve shown in Figure 5.36 at α = 0.3, bc bc E U bc bc E 10, ∞...... 117

xiv E U E 5.38 Enlarged pictures of Closed Contour-3 ( 9 , 10, , E E U E U E 10 (unstable), 10), Closed Contour-4 ( 10, ∞, E∞(unstable)); the equilibrium state E10 from two branches coalesce and form in to a single branch with change in α...... 119

E U 5.39 Enlarged pictures of G∗ as function of β, Closed Contour-3 ( 9 , E E E U E U 10, 10 (unstable), 10) , Closed Contour-4 ( 10, E∞, E∞ (unstable)); the equilibrium state E10 from two branches coalesce and form in to a single branch with change in α...... 120

E U E 5.40 Enlarged view of Closed Contour-2 ( 8 , 9, E E U E U 9(unstable), 9 ), Closed Contour-3 ( 9 , E E E U E E 10, 10(unstable), 10, ∞, ∞(unstable)), E U with increase in α the branches 9 on both the contours start merging. 121 E U E E 5.41 Enlarged view of Closed Contour-1 ( 7 , 8, 8 (unstable), E U ) and Closed Contour-2 ( E U , E , E 8 bc bc 8 9 9 E U E E E U E (unstable), 9 , 10, 10 (unstable), 10, ∞, E E U ∞ (unstable)), with increase in α the branches 8 on both the contours start merging...... 122

E U E U 5.42 Enlarged view of Open Contour ( 6 , 7 ) and Closed E U E E E U Contour-1 ( 7 , 8, 8 (unstable), 8 ) and Closed E U E E E U E E Contour-2 ( 8 , 9, 9(unstable), 9 , 10, 10 E U E E (unstable), 10, ∞, ∞ (unstable)), with increase in α E U the branches 7 on both the contours start merging...... 123

5.43 (a) Global Coupling Force G∗, (b) Global Self-Consistency Variables (V,W ), as a function of β, is now a single continuous single curve at α = 0.86...... 124

5.44 Equilibrium distributions at (α , β) = (0.2 , 4.5), where they look E 1 E 2 similar geometrically each named as 8 (stable), 8 (unstable)...... 126 E 1 5.45 Spectrum plots of distributions 8 (stable) shown in (a) and E 2 8 (unstable) shown in (b), when (α , β) = (0.2 , 4.5)...... 126 5.46 A curve separating stable and unstable regions on (α , β) plane for equilibrium E7 distribution, A and B are two discrete points of (α , β) for which the distribution is stable and unstable respectively. . . 127 5.47 Equilibrium distributions at for two different coupling points at A when (α , β) = (1.6 , 1.8) is shown in (a) and at B when (α , β) = (1.9 , 3.0) is shown in (b)...... 128

xv E 1 5.48 Spectrum plot of distribution 8 (stable) is in Figure(a) and for E 2 distribution 8 is in (b), this corresponds two different coupling parameters A and B inside and outside of the stable region as shown in Figure 5.46...... 129 5.49 The curves separating stable and unstable regions on α , β plane for different equilibrium states E8 , E9 , E10 , and E∞ are shown here. . . . . 131

5.50 Equilibrium distribution E∞ at (a) β = 20, (b) β = 40, when α = 0.45. . 132

5.51 Eigenvalues E∞ at (a) β = 20, (b) β = 40, when α = 0.45, indicates these equilibrium distributions are stable...... 132

xvi CHAPTER I

INTRODUCTION

All modern electronic devices for both industrial and household applications have been scaled down in size and their performance has been improved over the last few decades, this can be attributed to effective packaging and fabrication techniques in the semiconductor industry. The functionality of RLC (Resistor Inductor Capaci- tor) circuits can be used for sensing and signal processing when they can allow some mechanical motion, these types of devices which have electric/mechanical or both an input/output are loosely termed as Micro-Electro-Mechanical Systems (MEMS) devices [6–9]. Microdevices offer many advantages in terms of power consumption, sensitivity and cost compared to their macroscale counterparts. MEMS devices were commercially utilized in the making of ink-jet print heads, accelerometers, gyroscopes, pressure sensors, and magnetometers [10, 11]. High throughput, good frequency accuracy and system integration coupled with miniaturization make them an excellent choice in sensing, pattern generation, and communication applications. MEMS-based resonators, herewith referred to as microresonators are components that respond at the well-defined frequency when actuated by an external excitation [12–14].

The majority of modern microresonators utilize harmonically-forced, linear resonators, which exhibit a Lorentzian frequency response curve. The major proper-

1 ties that define a microresonator are the resonant frequency and quality factor (an inverse measure of damping). This simplistic approach has led to an easier design, fabrication, and integration of linear microresonators, which is currently a multibillion- dollar industry, but at the cost of flexibility and realizing the true potential of these micro-devices. Many MEMS devices feature inherent nonlinearities, although nonlinear frequency response structures traditionally have been avoided within the MEMS research community. Nonlinearities in microresonators generally arise from three distinct sources, large structural deformations, displacement-dependent excitations, and tip/sample interaction potentials, such as the Lennard-Jones potential, which arise in atomic force microscopy [15–17]. The nonlinear microresonators have re- ceived meaningful attention only recently, starting with investigations of microresonators emphasized force harmonic oscillators, typically planar comb-driven devices, with non-linearities arising from large elastic deformations. These devices exhibited classical Duffing frequency response structures, which offered several benefits like lower sensitivity to damping, mechanical gain near the onset of bi-stability, and high inter-modulation gain when the system is operated close to the bifurcation. Though microresonators based on nonlinear frequency response structures are more difficult to design and analyze than their linear counterparts, they offer a degree of tunability unattainable with a linear device and have the potential to exhibit superior performance metrics [18]. The development of nanoresonators requires a better understanding of nonlinear response since they have limited linear dynamic array [19, 20].

Single resonator MEMS devices are used in chemical, inertial and acoustic

2 sensing, atomic force based microscopy, computing, and radio-frequency communications, but they suffer from severe throughput limitations i.e., the amount ofin- formation they can sense/process. This limitation was initially overcome by exploiting massive parallelization of large arrays of uncoupled resonators. Since the signal processing and hardware requirements attendant to this approach largely negate its practical utility, in general, most of the sensors were designed for signal input and output for each channel of information contained within the system. Recent research in such systems indicated that constraints on the throughput can be addressed by utilizing the collective behaviors like synchronization, localization, spatial confinement by coupling them as array of micro/nanoresonators [21–27]. This can greatly enhance performance metrics like sensitivity, selectivity, and attenuation. Different technical challenges are encountered, in the design and analysis of these dynamical systems based on complex global behavior due to the inherent sensitivity to parameter variations and noise. Moreover, phase-based mathematical models [28–30] that were developed previously are not practically suited to address the intricacies involved with the latest design requirements of MEMS devices. This document is an attempt to develop analysis methods to study the dynamics of linear and nonlinear dynamics of large arrays of coupled resonators operating under the influence of parametric uncertainty.

3 1.1 Single Resonator Dynamics

Resonators are actuated using transverse/ axial loads either using electrostatic, piezoelectric or electromagnetic excitations, with boundary conditions of free-free, free- fixed, fixed-fixed, etc combinations. Here a spatiotemporal model of a typicalres- onator is developed using classical beam theory, this distributed model is subsequently reduced to a comparatively-simpler lumped mass model [1,31]. This model can both serve a parametrically-excited and externally excited system of resonators.

1.1.1 Modeling of Resonator

Large deformation of a structure does not necessarily mean the presence of large strains, under large rigid-body rotations, structures like cantilever beams undergo large deformations but small strains. Since for majority of the cases as tip deflection of cantilevered beam is used for sensing the maximum deflection, MEMS resonator for the case under considered to be a cantilever beam in Figure 1.1. The cantilevered beam undergoes longitudinal, transverse and angular displacements of the differential beam element, u, v, and ψ respectively.

Assuming that the microbeam resonators of interest are uniform and have negligible rotational inertias (the assumption is justified by the slenderness of the microbeam systems of interest), then the specific Lagrangian of the resonator can be approximated as

1 1 ′ L = m (u ˙ 2 +v ˙ 2) − D (ψ )2, (1.1) 2 2

4 Figure 1.1: Schematic diagram of beam element used as resonator, here u, v, and

ψ are longitudinal, transverse and angular displacements of the differential beam element [1].

where m and D are the specific mass and flexural stiffness of the beam respectively, u, v, and ψ are as defined in Figure 1.1. Further assuming that the neutral axisof beam is inextensible, following condition is obtained

′ ′ (1 + u )2 + (v )2 = 1 (1.2)

′ v tan ψ = (1.3) 1 + u′

′ ∂ and () = ∂s is the partial derivative with respect to space, while the partial derivative

˙ ∂ with respect to time is denoted as () = ∂t . Using the extended Hamilton’s principle for arriving at the governing equations

∫ ∫ t2 l [ 1 ′ ′ ] δH = δ L − λ[1 − (1 + u )2 − (v )2] ds dt 2 t1 0 ∫ ∫ t2 l + (Quδu + Qv δv)ds dt, (1.4) t1 0

5 here Qu, Qv represents planar, non-conservative forces in the u and v directions, and λ is the Lagrange multiplier for in-extensibility constraint. In view of brevity the above

Hamilton’s equation and considering viscous damping cv is reduced to the deflection in v(s, t),

iv ˜ ′ ′ ′′ ′ ′ m v¨ + cv v˙ + E I v = f(s, t) − EI[v (v v ) ] . (1.5) where E and I are the Young’s modulus and moment of inertia respectively, and f(s, t) is external excitation, it is subject to following boundary conditions,

′ ′′ ′′′ v(0, t) = 0 , v (0, t) = 0 , v (l, t) = 0 , v (l, t) = 0. (1.6)

1.1.2 Reduced order modeling

In order to facilitate nonlinear analysis and ultimately predictive design, the governing partial differential equation presented Eq. (1.5) can be reduced to an ordinary differential equation by expanding the dynamic variable

∑∞ v(s, t) = zi(t)ψi(s), (1.7) i=1 where ϕi(s) are undamped mode shapes of cantilever beam, they can be expressed here by

[ ( )] 1 ris ris cos ri + cosh ri ris ris ψi(s) = √ cosh − cos + sin − sinh , (1.8) (l) l l sin ri + sinh ri l l

the variable r1 in Eq. (1.8) can be identified with the root of

1 + cos r1 cosh r1 = 0. (1.9)

6 Since the effects of primary modes is established to be dominant fromex- perimental studies and there is little improvement in accuracy by including higher order harmonics, only the first mode is considered for the expansion and reduced non-dimensionalized equation can be written as

z¨ + 2 ζ ω z˙ + ω2 z + γ z3 = f(t). (1.10)

The new non-dimensionalized variables of Eq. (1.10) √ ∫ ∫ l l 2 EI cv 3EI ′ 2 ′′ 2 ˜ ω = r1 4 , ζ = , γ = ψ1(s) ψ1 (s) ds , f = f ψ1(s)ds. (1.11) ml 2mω m 0 0

1.1.3 Solution of Duffing equation

Eq. (1.10) represents the Duffing resonator excited by an external force, it is asimple extension of a damped harmonic oscillator with additional non-linear stiffness. When an external excitation is a periodic force with a frequency close to the natural excitation a sudden transition in the response of the system can be observed, this can further be used for sensing purposes. The Duffing equation mentioned here can have closed form solution under the assumptions of small damping, weak non-linearity, and small periodic force, these parameters transform to,

ζ −→ ϵζ , γ −→ ϵγ and f(t) −→ ϵ f(t). (1.12)

Assuming the external excitation to be harmonic force, with frequency Ω and magnitude F0 is f(t) = F0 sin(Ω t). Since it is assumed that excitation frequency is close to natural frequency of resonator Ω = ω + ϵ σ (here σ indicates the frequency

7 detuning). The transformed equation can be written as,

2 3 z¨ + ω z = ϵ(F0 sin(ω + ϵ σ)t − 2 ζ ω z˙ − γ z ). (1.13)

Multiple scales approach is applied to this system with the identification of time scales τ = Ωt (fast time) and η = ϵt (slow time), so that

d ∂ ∂ = Ω + ϵ , dt ∂τ ∂η (1.14) d2 ∂2 ∂2 = Ω2 + 2ϵΩ + O(ϵ2), dt2 ∂τ 2 ∂τ ∂η while the response of each resonator is expanded as a series in ϵ as

2 z(τ, η) = z0(τ, η) + ϵ z1(τ, η) + O(ϵ ). (1.15)

Substitution of this expansion and the multiple time scales into Eq. (1.13), and col- lecting terms with coefficients of ϵi yields

O(ϵ0):

∂2z 0 + z = 0, (1.16a) ∂τ 2 0

O(ϵ1):

∂2z ∂z 2 1 − 2 0 − 3 Ω ( 2 + z1) = [F sin τ 2 ζ Ω γ z0 ∂τ ∂τ (1.16b) ∂2z + 2σΩ z − 2 Ω 0 ] 0 ∂τ η

The lowest order response of z(τ, η) is

z0(τ, η) = R(η) sin(τ + ϕ(η)), (1.17)

8 π 10.0 2 (a) (b) 8.0 0 ) 6.0 ) π σ −σ ( ( 2 ϕ R 4.0 −π 2.0 − 3 π 0 2 −5 0 5 −5.0 0 5.0 σ σ

Figure 1.2: F0 = 2.0 , ζ = 0.1 , γ = 0.1 a)Amplitude R(σ) vs σ b) Phase ϕ(σ) vs σ

Introducing z0 into next order of the approximation leads to secular terms in the system—for uniformly valid solutions these must be eliminated which provide equations on the slow time scale η that determine the evolution of (R(η), ϕ(η))

dR F = −ζ R + 0 sin ϕ, (1.18a) dη 2ω dϕ 3γ R3 F R = σ R − + 0 cos ϕ. (1.18b) dη 8Ω 2ω

The steady state solution for amplitude and phase are shown in Figure 1.2 as a function of detuning, it is to observed that peak amplitude of the resonator doesn’t occur at resonant frequency (σ = 0) but away from it, also there exists multivalued- ness in the response as σ changes, this is attributed to non-linearity present in the

Eqs . (1.18), there is also a jump in both phase and amplitude after the peak value is reached, which can be amply leveraged to sense a frequency change.

9 10 0 ) 5 /2 ) R( (

0 - -20 -10 0 10 20 -20 -10 0 10 20

Figure 1.3: F0 = 2.0 , ζ = 0.1 a)Amplitude R(σ) vs σ b) Phase ϕ(σ) vs σ, γ =

−0.5, γ = −0.1, γ = 0, γ = 0.1, γ = 0.5.

1.1.4 Affect of parameters on system behavior

Figure 1.3 shows the influence of non-linear stiffness coefficient γ on the behavior of resonator. The peak amplitude value remains the same but it occurs at different detuning values. The negative value of non-linear stiffness shifts the occurrence of peak amplitude at the negative value of detuning, this is described as softening. The reverse phenomenon in which the shift is towards the positive value of detuning for the positive value of gamma is hardening.

The effect of external excitation on frequency response of the Duffing resonator is represented in the Figure 1.4, as the actuating force increases the detuning value at which peak amplitude shifts to right of σ = 0 axis, the backbone

3γR2 curve(σ = 8ω ) which is the locus of peak amplitudes at different excitations is also shown, it is to be noted that multivalued of response for the resonator depends on the excitation, when it is less a σ has unique value.

10 20

15 ) 10 R(

0 -10 -5 0 5 10

Figure 1.4: γ = 0.1 , ζ = 0.1 Frequency response R(σ) vs σ at F0 = 0.5,

F0 = 1.0, F0 = 3.0, backbone curve representing the locus of peak

3γR2 amplitudes(σ = 8ω )

Figure 1.5 shows the effect of damping on the frequency response ofthe

Duffing resonator when the damping is absent the peak amplitude is infinite andtwo branches of asymptotically meeting at infinity. Increasing the damping would reduce the amplitude and also shifts the peak amplitude nearer to σ = 0.

Hence the Duffing resonator which is an extension of a simple harmonic system with a non-linear stiffness term offers a jump in the response when operated near the resonant frequency which can be utilized for sensing purposes. The location of peak amplitude depends on the parameters of the resonator discussed above.

1.2 Coupled resonators

In the previous section, a continuous model of a MEMS resonator is taken and reduced to a single degree of freedom stiffness using the modal expansion method, which can

11 20

15 ) 10 R(

0 -10 0 10

Figure 1.5: γ = 0.1 ,F0 = 2.0 Frequency response R(σ) vs σ at ζ = 0.05,

ζ = 0.1, ζ = 0.2, ζ = 0.5.

Figure 1.6: Coupled system of resonators with 2 degree of freedom systems

be used for analysis. Here two such identical resonators are weakly coupled to leverage sensitivity, performance, and throughput of the system. This vibration localization in closely detuned systems leads to enhanced parametric sensitivity using a shift in frequency, which is demonstrated in this section. In Figure 1.6 two non-linear resonators with its parameters are connected by a linear spring and a damper is actuated by external excitation.

These two resonators are coupled by a weak spring(kg) and damping element(cg),

12 the governing differential equation of the coupled system are,

3 − M1 z¨1 + c1 z˙1 + K1 z1 + K1nl z1 + cg (z ˙1 z˙2)

+ Kg(z1 − z2) = f1(t) , (1.19a)

3 − M2 z¨2 + c2 z˙2 + K2 z2 + K2nl z2 + cg (z ˙2 z˙1)

+ Kg(z2 − z1) = f2(t). (1.19b)

Under the assumptions of weak non-linearity, damping, coupling also sinusoidal excitation near to the resonant frequency, the non-dimensionalized equations in terms of new variables can be written as,

3 2 z¨1 + 2 ζ1 ω1 z˙1 + γ1 z1 + ω1 z1

= ϵ [ F0 sin(Ω t) − αg (z ˙1 − z˙2) − βg (z1 − z2)] , (1.20a)

3 2 z¨2 + 2 ζ2 ω2 z˙2 + γ2 z2 + ω2 z2

= ϵ [ F0 sin(Ω t) − αg (z ˙2 − z˙1) − βg (z2 − z1)] . (1.20b)

Using the multiple-time scales and scaling the coupling terms, damping and excitations by small order parameter ϵ as discussed previously and O(ϵ0),

z1(τ, η) = A1(η) sin(τ) + B1(η) cos(τ) , (1.21)

z2(τ, η) = A2(η) sin(τ) + B2(η) cos(τ). (1.22)

13 Subsequent slow-flow equations obtained by equating the secular term yields first order differential equations

∂A −1[ B (A2 + B2) 1 = 2 ζΩ2 A + 2 σ Ω B + 3γ 1 1 1 ∂η 2Ω 1 1 1 4 ] + Ω αg (A1 − A2) + βg (B1 − B2) , (1.23a)

∂B −1[ A (A2 + B2) 1 = F − 2 ζΩ2 A + 2 σ Ω B − 3γ 1 1 1 ∂η 2Ω 1 1 1 4 ] − Ω βg (B2 − B1) − αg (A1 − A2) , (1.23b)

∂A −1[ B (A2 + B2) 2 = 2 ζΩ2 A + 2 σ Ω B + 3γ 2 2 2 ∂η 2Ω 2 2 2 4 ] + Ω αg (A2 − A1) + βg (B2 − B1) , (1.23c)

∂B −1[ A (A2 + B2) 2 = F − 2 ζΩ2 A + 2 σ Ω B − 3 γ 2 2 2 ∂η 2Ω 2 2 2 4 ] − Ω αg (B1 − B2) − βg (A2 − A1) . (1.23d)

2 2 1 The system variables in Eqs. (1.23) can be reduced to R1 = (A1 + B1 ) 2 ,

2 2 1 R2 = (A2 + B2 ) 2 , which represents the amplitude response of the resonator 1 and

2 respectively. Frequency response for 2-degree of freedom system in the absence of non-linearity(γ = 0) is shown in Figure 1.7, this reduces to two simple harmonic resonators coupled by damping unit with αg and stiffness element kg. It is assumed that two resonators have natural frequencies 1 + ϵσ1 and 1 + ϵσ2( numerical values of σ1 = −5.0, σ2 = 5.0, ϵ = 1e − 2). When the resonators are excited by external 14 harmonic force at frequency range near the resonance frequencies, the maximum response is observed near the natural frequency of individual resonators. In the absence of coupling αg = 0 , βg = 0 response is seen in Figure 1.7(a) where the response of resonator 1 and 2 is maximum at their respective undamped natural frequencies and response is inversely proportional to the damping. With the change in stiffness coupling constant βg to 0.5 in Figure 1.7(b), resonators are coupled and when Ω ≈ 0.95 the system is near its primary natural frequency, where both resonator

1 and 2 move in phase and amplitude of 1 is dominant compared to 2. Similarly when Ω ≈ 1.05 then resonators 1 and 2 are out of phase with each other, where amplitude response of 2 is dominant compared to that of 1. With further increase in coupling strength as seen in Figure 1.7(c) and Figure 1.7(d) the primary mode shifts away(decreases) from its mean excitation frequency, similarly, secondary mode moves closer to mean excitation. It is also to be noted that with an increase in coupling stiffness βg amplitude response of resonator 1 decreases and resonator 2 increases near the primary mode, amplitude of both resonators 1 and 2 increases near secondary mode. There is also antiresonance observed with an increase in coupling strength.

Figure 1.8 shows the effect of coupled damping element on frequency response characteristics of the system at stiffness coupling of βg = 0.5. With increase in βg from 0 to 0.5 the amplitude response at both primary and modes are almost scaled down by half as seen from the responses in Figure 1.8(a) and Figure 1.8(b). Further increase in βg leads to almost decoupling the amplitude of resonator 2 near primary mode and resonator 1 near secondary mode indicating the modal damping due to

15 5.0 5.0 (a) (b) 4.0 4.0

) 3.0 ) 3.0 s s ( (

R 2.0 R 2.0 1.0 1.0 0 0 0.8 0.9 1.0 1.1 1.2 0.8 0.9 1.0 1.1 1.2 Ω Ω 5.0 5.0 (c) (d) 4.0 4.0

) 3.0 ) 3.0 s s ( (

R 2.0 R 2.0 1.0 1.0 0 0 0.8 0.9 1.0 1.1 1.2 0.8 0.9 1.0 1.1 1.2 Ω Ω

Figure 1.7: F0 = 2.0 , ζ1 = ζ2 = 0.25 , αg = 0 R(1)( σ1 = −5.0), R(2)(

σ2 = 5.0) vs Ω (excitation frequency) for (a) βg = 0 , (b) βg = 0.5 , (c) βg = 5.0 and

(d) βg = 10.0 .

16 5.0 5.0 (a) (b) 4.0 4.0

) 3.0 ) 3.0 s s ( (

R 2.0 R 2.0 1.0 1.0 0 0 0.8 0.9 1.0 1.1 1.2 0.8 0.9 1.0 1.1 1.2 Ω Ω 5.0 5.0 (c) (d) 4.0 4.0

) 3.0 ) 3.0 s s ( (

R 2.0 R 2.0 1.0 1.0 0 0 0.8 0.9 1.0 1.1 1.2 0.8 0.9 1.0 1.1 1.2 Ω Ω

Figure 1.8: F0 = 2.0 , ζ1 = ζ2 = 0.25 , βg = 0.5 R(1)( σ1 = −5.0), R(2)(

σ2 = 5.0) vs Ω (excitation frequency) for (a) αg = 0 , (b) αg = 0.5 , (c) αg = 2.0 and

(d) αg = 10.0 .

the coupling, this is indicated in Figure 1.8(c) and Figure 1.8(d). The effect of cubic nonlinearity on a coupled system as the actuating frequency is varied can be seen in

Fig 1.9.

The two possible Non-linear Normal Modes(NNM) can be distinctly seen from these response curves. Just like a single resonator case was seen in the previous section even two degrees of freedom system shows a sudden change in the amplitude response at different non-linear stiffness constant (γ) indicating multivalued response functions.

The cutoff or jump frequency is determined by the strength of non-linearity, foragiven

γ at both primary and secondary modes the jump in response for resonator 1 and 2 occurs at same cut-off. This phenomenon can be utilized for sensing applications as

17 output from two resonators indicating the sudden drop in response when converted to some secondary measuring variable like voltage change is more robust compared to output from a single resonator. In Figure 1.9(c) and Figure 1.9(d) it can be seen that the width of anti-resonance for both resonators 1 and 2 increases with an increase in coupling strength.

4.0 4.0 (a) (b)

b 3.0 3.0 b b b b b b b

b b ) ) b b b s s b b b b ( 2.0 ( 2.0 b b b b

R b b R b b b b b b b b b b b b b b b b b b b b b b b b 1.0 b b 1.0 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb b b b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b 0 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb 0 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb 0.8 0.9 1.0 1.1 1.2 0.8 0.9 1.0 1.1 1.2 Ω Ω 4.0 4.0 (c) (d)

3.0 b 3.0 b b b b b b b b b b b b b b ) b ) b b b b b b b

s b s b b b b b b b ( b ( b b 2.0 b 2.0 b b b b b b b b b b b R b R b b b b b b b b b b b b b b b b b b b b b b b b b b b b 1.0 b 1.0 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb b b b b b b b b b b b b b b b bb bb bb b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b 0 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb 0 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb bb 0.8 0.9 1.0 1.1 1.2 0.8 0.9 1.0 1.1 1.2 Ω Ω

Figure 1.9: F0 = 2.0 , ζ1 = ζ2 = 0.25 , βg = 2.0 , αg = 0.2 R(1)( b b σ1 = −5.0),

R(2)( b b σ2 = 5.0) vs Ω (excitation frequency) for (a) γ = 0 , (b) γ = 0.08 , (c) γ =

0.5 and (d) γ = 2.0 .

Hence using an additional resonator in place of a single resonator enhances the capability of it being used for sensing purposes. In mechanical systems and structures, each sub-unit is connected to the nearest-neighbor, which is termed as

18 local coupling. Here coupling strength between resonators and uncoupled natural frequencies affect the system behavior. Several interesting phenomena like confined attenuation, group attenuation, and group resonance have been observed in the case of globally connected systems available in the literature. A large array of MEMS resonators connected globally had also been demonstrated and the coupled dynamics of such systems need to be properly analyzed to avail the benefits mentioned above.

1.3 Organization of the Thesis

This work is aimed at providing a thorough understanding of the dynamics of large array resonators which are globally coupled, here second order differential equations which have non-linear stiffness, reactive coupling, dissipative coupling governing the

N coupled resonators are taken as a preliminary point. From previously discussed techniques in literature, which includes using the mean-field method of equilibrium statistical mechanics to convert these differential equations into a single non-linear integro-differential equation has been exploited. Finally, the multiple time scale analysis technique is used to obtain the final first order equations also called slow-flow equations, which can determine transient and steady-state dynamics of the system.

Spectrum analysis techniques are used to determine the stability of multiple equilibrium states.

Chapter 1 of this report contains, basic introduction of MEMS resonators, preliminary deductions of governing equation of an isolated resonator from a continuous cantilever model using mode expansion is shown. Analytical treatment of 19 Duffing resonator operating near resonance frequency is done, the effect ofdifferent parameters on the response characteristics is explained. A two resonator coupled system is analyzed with the help of multiple time scale analysis, frequency response curves are used to study the effect of coupling parameters and non-linear stiffness.

The first part of Chapter 2 contains a brief description of MEMS devices, different classification in MEMS resonators, actuation, sensing mechanisms usedin them. Chronological development of coupled resonators starting with their utility as bandpass filters, comb-driven resonators, etc, it also talks of advances made forboth linear and non-linear coupled resonators. The second part emphasizes on mathematical developments used in modeling self-sustained oscillations due to the presence of coupling, solution procedures that were used for solving large population models like

Winfree and Kuramoto are discussed. Unfortunately, these methods are ill-suited for many practical problems, which feature both amplitude and phase dynamics, inherent element-level parameter variation (mistuning), nonlinearity, and/or noise.

Chapter 3 focuses on developing a method to solve coupled systems in which the number of individual resonators is large, extending as far as the continuum limit corresponding to an infinite number of resonators. The utility of this analytical methodology is explored through the analysis of system of N non-identical resonators with global coupling, including both reactive and dissipative components, physically motivated by an electromagnetically-transduced micro resonator array. The method of multiple time scales has been used to for obtaining the slow flow equations that can predict both dynamic and steady-state behavior of coupled resonator system. The

20 equations to determine the stability of equilibrium states has been derived using the concepts of discrete spectrum.

The monostable equilibrium distributions which exist for system of resonators that have low non-linearity or purely linear systems are discussed in the Chapter 4.

Effect of global self-consistency variables which essentially decides the response of the system and its dependence on the coupling parameters along with the number of resonators N has been discussed. The behavior of self-consistency variables in case of random and normal distributions is also clearly explained.

In Chapter 5 the multiple equilibrium states for a non-linear system has been analyzed, the types of bifurcations that results in the multiple equilibrium states has been highlighted. The interesting behavior of self-consistency variables with change in the coupling parameters leads to bi-stability of resonators locally, and further increase of the same also results in change of its behavior at global level, this phenomenon is clearly explained using global coupling force and multiple distributions resulting from it.

Chapter 6 Outlines the summary of all the conclusions made in this study and the prospective future work of this research.

21 CHAPTER II

LITERATURE

Microelectromechanical systems (MEMS) are the devices that use Integrated Circuits

(IC) technology and mechanical motion for achieving their desired functionality and purpose. The earliest advances in the fabrication of MEMS-like devices were shown by H. C. Nathanson [32] in the fabrication of resonant gate transistor and Newell [6] in making of his tunistor, another significant milestone in this field had come when Kurt

Peterson demonstrated the feasibility of silicon for mechanical device fabrication [33].

In the last three decades application and usage of MEMS sensors/devices have grown leaps and bounds in various fields including communications, sensing, physics and technology [34–42].

Microresonators are the special class of MEMS devices which show resonant behavior or large amplitude vibrations when actuated at specific frequencies. They are very simple in their design and consist of a few common elements such as beams, lumped masses, etc [43]. These MEMS-based resonators are classified based on the actuating and sensing methods used for its operation, this is collectively termed as transduction. Different types of actuation methods are currently available, ofwhich electrostatic, piezoelectric, piezoresistive, and electromagnetic are mostly in use. A combination of them can be used for actuation and sensing purposes, each of these

22 has their advantages and disadvantages in their implementation [7, 44–46]. These microresonators are subjected to either parametric or harmonic excitations to be used as mass sensors, bandpass filters, signal-processors and concentration, and temperature sensors [47–50]. Investigations in earlier times focused on electrostatically actuated and sensed devices, specially using parallel-plate capacitive structures, with advancement of microfabrication technologies matured, multiple linear devices based on alternative transduction mechanisms emerged, including systems utilizing piezoelectric [51,52], electromagnetic [53,54], thermal [55,56], and optical [57,58] actuation and/or sensing elements. The systems that were developed previously for RF signal filtering and resonant mass sensing applications were adapted for use in inertial sensing, pressure sensing, radiation sensing, magnetic field sensing, resonant pumping, and optical scanning system [7,59–61]. Most of the work in initial years was with linear resonators, which have Lorentzian nature of response functions and large quality factor Q (lesser the damping larger is the quality factor). As Q increases, the frequency bandwidth over which the microresonator has a response amplitude of similar magnitude to its maximum value becomes narrower [62]. So, the designers had to optimize between the response and bandwidth in the case of linear resonators. Their fabrication, design, and integration have reached a mature stage and abundantly used in various disciplines of science and engineering. Microresonators have many sources of nonlinearities due to large structural deformations, displacement-dependent excitations, and tip interactions potentials, such as the Lennard-Jones potential, which arise in atomic force microscopy [63]. Previous studies of nonlinear microresonators

23 emphasized on forced harmonic oscillators, typically planar comb-driven devices, with nonlinearities arising from large elastic deformations. Such devices exhibited classical

Duffing frequency response structures, which offer many additional benefits likelower sensitivity to damping but were deemed to be inferior to their Lorentzian counterparts for most applications [64]. Only recently the study of the nonlinear response of microsystems undergoing large elastic deformations in the presence of multi-physics excitations has nonlinear microresonators garnered serious attention [65–68]. The microresonators based on nonlinear frequency response structures are more difficult to design and analyze than their linear counterparts, they offer a higher degree of tunability compared to a linear device and have the potential to exhibit superior performance metrics.

2.1 Coupled Resonators

Coupled resonators have garnered a lot of interest since the seminal work of Lin et al in early-1990s on bandpass filters [2]. The researchers had demonstrated design and fabrication techniques of parallel and series filters using polysilicon microstructures with suspension beams having a 2µm square section as shown in Figure 2.1.

Their work targeted to use the chains of planar, comb-driven resonators, coupled through common elastic flexure, to mimic the behavior of LC ladder filters and initial mechanical analogs [43, 69]. They have also used weakly-coupled micromechanical resonators to obtain frequency response functions with narrow, well-defined, multi-resonance passbands. In subsequent years, Nguyen and others built upon the 24 Figure 2.1: SEM image of two resonant filter [2], series-coupled resonator deigned for operation at atmospheric pressure.

25 Figure 2.2: SEM image of polysilicon fabricated micromechanical disk-arrays [3],a)

Top-view b) Iso-view image.

success of this initial work by developing a variety of new devices incorporating alternative resonator geometries [3, 70–77]. Figure 2.2 shows a disk composite-array structure that is placed in constituent disks in a triangular pattern around a mechanical output disk, while coupling them with half-wavelength links, in order to accentuate a desired resonant mode while suppressing the unwanted modes.

Different coupling mechanisms implementing high order mechanically-coupled ultra-small bandwidth micromechanical filter arrays without using discrete coupling elements were tried [4, 78–80]. Adjacent resonators are coupled by elastic deformation at the support and filter bandwidth is defined by between resonating elements,

26 Figure 2.3: SEM image of four 500µm×4µm clamped-clamped array beams with 9µm spacing [4], through-support coupling was extended to large arrays simply placing beams adjacent to each other.

SEM image of four 500µm × 4µm clamped-clamped array beams with 9µm spacing is shown in Figure 2.3 higher-dimensional arrays [81–84], and frequency-mistuned subsystems [81]. The work referenced in [81] highlighted on relationship between classical vibration localization that results in spatial confinement of energy in the coupled systems with structural mistunings. This work subsequently led to exploiting vibration localization, particularly in the areas of mass sensing [26,27,85,86] and mechanics based on Fourier decomposition [87, 88].

Though most of the research related to coupled micro/nanoresonator arrays has focused in the linear domain, some more recent works have explored the potential of coupled dynamical systems utilizing nonlinear elements, nonlinear coupling

27 and the complex global behaviors that can be known from these effects. Hammad et al [88–93] detailed the development of refined nonlinear models for electro statically- actuated, elastically-coupled signal filters similar in design to those originally proposed by [76]. Buks, Roukes, Lifshitz and Cross had contributed electrostatically- coupled micro/nanoresonator arrays [94–97]. The earliest paper in this series by Buks and Roukes [94] to study the mechanical properties of doubly clamped beams of microresonator array made of Au under application of electrostatic coupling driven near parametric resonance, the effects of nonlinearity on the dynamics of thesys- tem were witnessed. Lifshitz and Cross addressed the collective response in similar systems using both analytical and numerical techniques for an array of coupled resonators [22,97–100]. Buks and Roukes [101] approached their 67-element array problem from continuous limit and transformed it to a spatially-discrete problem. This approach was useful in bifurcation analysis [102] in the proximity of the parametric resonance and opened the problem to predictive system design. Another important research area focused on micro resonator arrays is intrinsic localize modes (ILMs), this phenomenon is due to the presence of strong nonlinearity, rather than structural mistuning [103–107]. Sato et al has demonstrated that ILMs exhibited by periodic array of identical microresonator was provided by a strong mechanical nonlinearity.

Sabater et al [108] worked on modeling, analysis, predictive design, and control of self-excited oscillators, and associated arrays, observed in electromagnetically- actuated microbeams. They also worked on characterization of nonlinear behaviors arising in isolated oscillators and small arrays of nearly-identical, mutually-coupled

28 oscillators [5], by primarily focusing on modeling, analysis and experimental characterization of the nonlinear response in electromagnetically transduced microcantilevers using inductive and resistive coupling between input and output ports of the devices. In Figure 2.4, a microcantilever can be seen, it has a current-carrying metal- lic wire loop and is placed in a permanent magnetic field and an alternating current is supplied, this device is actuated by Lorentz forces arising of the magnetic field.

These vibrations, in turn, induce an electromotive force, which can be used for correlating with the dynamic response of the device. Development of coupled resonators arrays in the non-linear domain has not only improved sensitivity in mass sensing applications [85, 86] but also its potential application in new research domains like neural computing [109].

2.2 Mathematical models

Modeling of coupled systems have garnered a lot of interest since a very long time, first recognized effort dates back to Christiaan Huygens on his mention of synchronous clocks in 1665 [110], after the advent of calculus into scientific scene many researchers had immensely contributed to understanding this problem [111–117] with the help of coupled differential equations. If two systems are influencing the measured output parameters of the other then they are said to be coupled, the concept of coupling can be seen in various diverse fields like physics, engineering, mathematical biology [28, 118–122], but the scope of this report is more or less confined to coupled resonators/oscillator systems. 29 Figure 2.4: Scanning electron micrograph of a representative electromagnetically transduced microresonator. The device consists of an electrically and mechanically isolated silicon microcantilever and two Au/Cr wire loops, which follow the perimeter of the microcantilever [5].

30 The motion of coupled resonators is generally governed by second-order differential equations, when these equations involve purely linear terms they can be solved directly by various direct methods discussed herein [123–125] and also using different integral transforms (Laplace and Fourier). In the absence of external excitation and parametric uncertainty, only the complimentary function which is a solution to the homogeneous part exists. When there is time-dependent external excitation both complementary and particular solution plays an important role in the transient and steady-state response of the system, and also the equilibrium states are always unique for linearly coupled systems. When non-linear terms are involved in the differential equation it is not possible to solve using either transforms or regular integration techniques. Hence their behavior can be studied by rewriting higher-order (here 2nd order) differential equations in terms of first-order differential equations and charac- terizing the long-term behavior of the system at equilibrium points [126–128]. Making some realistic physical assumptions and appropriately scaling the system parameters the non-linear differential equations can be solved analytically using asymptotic methods as discussed in [129]. Whether it is a locally or globally coupled system, individual subsystems/units get influenced by similar units in the system through positive feedback affecting its motion, this, in turn, affects the dynamics of theentire system, here mathematical modeling of such systems are discussed.

The mathematical models of mutually-coupled electrical oscillators were developed by Vanderpol [130]. Even in the fields of biology where the firing of active neurons getting mutually synchronized or pacemaker cells spontaneously interacting

31 with each other so that they eventually function as a single oscillator is well known.

The biological oscillations are also witnessed by circadian rhythms and flashing fireflies are reported in [131,132], A.T.Winfree in his research have modeled eachofthis biological subsystems as oscillators differing slightly in their frequencies and are self- sustaining to collective rhythm due to the coupling present in the population, this is analogous to a mean-filed theory. The model proposed by Winfree [29],

( ∑N ) ˙ θi = ωi + X(θj) Z(θi) , i = 1,...,N (2.1) j=1 where θi denotes the phase of oscillator i and ωi its natural frequency. Each oscillator j exerts a phase-dependent influence X(θj) on all the others; the corresponding response of oscillator i depends on its phase θi, through the sensitivity function Z(θi). With the help of numerical simulations and analytical approximations, Winfree discovered that a population of oscillators could exhibit a time-dependent phase transition. When the distribution of natural frequencies is large compared to the coupling present between them, then the system behaves incoherently, with each oscillator moving at its natural frequency. As the distribution becomes narrower, the incoherence is observed until a certain threshold is crossed after which a small cluster of oscillators suddenly freezes into synchrony. Kuramoto inspired from Winfree model has proposed phase models [133–136], he used perturbative method of averaging to show that for any system of weakly coupled, nearly identical limit-cycle oscillators can be modeled as,

∑N ˙ θi = ωi + Γij(θj − θi), i = 1,...,N (2.2) j=1

32 The interaction functions Γij can be calculated as integrals involving certain terms from the original limit-cycle model [133]. Kuramoto recognized that the mean- field case should be the most tractable, Kuramoto reduced his model to thesimplest possible case of equally weighted, all to all, purely sinusoidal coupling:

K Γ (θ − θ ) = sin(θ − θ ) (2.3) ij j i N j i where K ≥ 0 is the coupling strength and the factor 1/N ensures that the model is well behaved as N → ∞. By further introducing two mean-field quantities r and ψ

Kuramoto has further reduced his model (details in [133]) to

˙ θi = ωi + K r sin(ψ − θi). (2.4)

Here r is coherence and ψ is mean phase, phase θi of the individual oscillator is pulled towards the mean phase ψ. The strength of coupling is proportional to the coherence r and this introduces a positive feedback loop between coupling and coherence. This makes oscillators in the population become more coherent, resulting in the growth of r which in turn increases effective coupling Kr , this further enhances more oscillators to get synchronized. If coherence is further increased the new set of oscillators gets recruited else it becomes a self-limiting case. It was shown that for K less than a certain threshold Kc, the oscillators act as if they were uncoupled, the phases become uniformly distributed around the circle, starting from an arbitrary initial condition.

Some problems which were left open from earlier Kuramoto’s work were of the linear stability, closed-form equation for critical coupling Kc and convergence results as N → ∞. Sakaguchi [137] extended Kuramoto model to allow rapid stochastic 33 fluctuations in the natural frequencies, and his model is written with independent white noises variable ξi(t) as

K ∑N θ˙ = ω + ξ + sin(θj − θ ), i = 1,...,N (2.5) i i i N i j=1

Strogatz and Mirollo [138] tried to address the problem of infinite by visual- izing for each frequency ω, there is a continuum of oscillators distributed along the circle and that distribution is characterized by a density function ρ(θ, t, ω). Where

ρ(θ, t, ω) dθ gives the fraction of oscillators of natural frequency ω which lie between

θ and θ + dθ at time t. Then the nonnegative ρ which is periodic in 2π satisfies the following normalization, ∫ 2π ρ(θ, t, ω) dθ = 1. (2.6) 0

The simplest state is uniform incoherent state is given by

1 ρ (θ, ω) = . 0 2π

Assuming that white noise satisfy following conditions,

⟨ ξi(t)⟩, ⟨ξi(s)ξj(t)⟩ = 2Dδijδ(s − t), (2.7) where D ≥ 0 is the noise strength and the angular brackets denote an average over realizations of the noise. Eq. (2.5) represents a system of coupled Langevin equations, the evolution of ρ(θ, t, ω) should satisfy Fokker-Planck equation [139] given by,

∂ρ ∂2ρ ∂ = D − (ρ v) (2.8) ∂t ∂θ2 ∂θ

34 Where v(θ, t, ω) is the velocity mentioned in,

v(θ, t, ω) = ω + K r sin(ψ − θ). (2.9)

Sakaguchi determined the closed form expression for critical coupling Kc,

[ ∫ ∞ ] D −1 Kc = 2 2 2 g(ω)dω , (2.10) −∞ D + ω where g(ω) is any distribution function representing natural frequencies oscillators,

+ in the limit as D → 0 the Kc = 2/πg(0) which was proved by Kuramoto for his model. Strogatz also extended his work to obtain linear stability of the incoherent state by introducing a perturbation about that point,

ρ(θ, t, ω) = ρ0(θ, t, ω) + ϵη(θ, t, ω). (2.11)

Assuming that exact solutions are complex conjugates of Fourier series in θ:

η(θ, t, ω) = c(t, ω) eiθ + c.c. + η ⊥ (θ, t, ω). (2.12)

κ ∑N θ˙ = ω + P (θ )R(θ ). (2.13) i i N j i j=1 where P (θ) = 1 + cos(θ) is a smooth pulse like function, R(θ) = −sin(θ) is qualitative shape of phase-response of biological oscillators [140,141]. These choices helped them to model biological systems like pulse-coupled biological oscillators, such as crickets, fireflies and heart pacemaker cells. They chose frequencies from evenly space interval

− 1 I = [1 γ , 1 + γ], hence the distribution function ended up being g(ω) = 2γ . In the limit as N → ∞, the Eq. (2.13) turns to be ∫ ∫ [ 2π 1+γ ] v(θ, t, ω) = ω − κ (1 + cos(θ)) p(θ, t, ω)g(ω) dω dθ sin(θ). (2.14) 0 1−γ 35 By solving this equation in closed form they were able to draw clear boundaries between locking, partial locking, partial death and incoherent states of resonators on plane of coupling strength (κ) vs width(γ). Quinn et al [142] used the model proposed in Eq 2.14 and modified it as follows, ∫ ∂Θ 1 (t, ν) = 1 + Γ ν − κ sin Θ(t, ν) [1 + cos Θ(t, µ)]h(µ)dµ, (2.15) ∂t −1 for all −1 ≤ ν ≤ 1. Here Θ(t, ν) denotes the phase at time t of an oscillator with natural frequency 1 + Γν. The mean natural frequency across the population has been set to unity, without any loss of generality. The parameter Γ represents the width of the frequency distribution, and −1 ≤ ν ≤ 1 is a normalized detuning, a measure of how far an oscillator’s natural frequency deviates from the population mean. The term with in the integral [1 + cos Θ(t, µ)] describes the influence of an oscillator with detuning µ, on the given oscillator with detuning ν. These influences are expressed with a weight h(µ), the probability density of natural frequencies across the population and it needs to satisfy normalization condition ∫ 1 h(µ)dµ = 1. (2.16) −1

This formulation allowed them to handle both discrete and continuous distributions of natural frequencies in a single framework. To represent the discrete case where oscillators have detunings νi, for i = 1,...N, then density function can be expressed as a sum of discrete delta functions as follows,

1 ∑N h(ν) = δ(ν − ν ). (2.17) N i j=1 36 Substituting the Eq. (2.17) in Eq. (2.15) again leads to a set of coupled differential equations. The remarkable feature of this approach is by choosing ν as a continuous indexing variable the problem of a coupled discrete differential equation can be converted to a single integro-differential equation and choice of density function helps easier shift from discrete domain to continuous space and vice-versa. The Eq. (2.15) was solved using Poincaré-Linstedt perturbation method [127] and appropriately scaling the different parameters to obtain asymptotic solutions.

Ott et al [143] obtained closed form solution for the nonlinear time evolution of the Kuramoto problem with a Lorentzian oscillator frequency distribution function.

Expanding population density function f(θ, ω, t) in a Fourier series in θ, { } [ ∑∞ ] inθ f = (g(ω)/2π) 1 + fn(ω, t)e + c.c. . (2.18) n=1

n Considering for class of functions fn(ω, t) = [α(ω, t)] , where |α(ω, t)| ≤ 1 to avoid divergence of the series. It turns out that density function is a geometric progression and the Kuramoto problem of large dimension was reduced to low dimensional behavior in phase plane r, θ. L.M.Childs and Strogatz et al [144] worked on Kuramoto model uncovered two main types of attractors, called forced entrainment and mutual entrainment using the ansatz suggested by E.Ott [143]. They could reduce an infinite-dimensional dynamics to a two-dimensional system with a complete bifurcation diagram. Using the similar ansatz as above, Keeffe et al [145] presented the stability diagram and calculated several of its bifurcation curves analytically, for both excitatory and inhibitory coupling. For a broad and pulsed coupling function, their

37 model exhibited bi-stability between steady states of constant high and low activity, for a narrower coupling function, they found that the bistable regime has continuous pulsations in activity.

The methods discussed so far are mathematical reductions to phase-only models or leverage system symmetries to predict and explain observed dynamical behaviors. Unfortunately, these methods are ill-suited for many practical problems, which feature both amplitude and phase dynamics, inherent element-level parameter variation (mistuning), nonlinearity, and/or noise. So a coupled microresonator array with reactive and/or dissipative coupling, together with element-level dynamics that contain stiffness nonlinearities under the influence of external excitation istaken up for developing the new analytical framework which is taken up for discussion in subsequent chapters by extending the approach suggested in [142].

38 CHAPTER III

ANALYTICAL MODELING OF LARGE ARRAY RESONATORS

The micro resonators which are one of the copious MEMS devices utilized in various discipline of engineering and science. Their applications have been discussed in previous chapter [27, 81, 85, 146] in detail. Coupled resonators have garnered a lot of attention in the recent times due to their high throughput, confined attenuation, group resonance, in this area of application there exist natural economies of scale, which allow for very large degree-of-freedom ensembles, accompanied by noise and parametric uncertainty [21,24,25,102,109,147]. The emergent dynamics of such systems are often avoided, yet with proper system design, the global dynamics can be exploited to provide system performance that cannot be realized with individual components. Accordingly, these systems offer a certain appeal for the traditional analysts, as well as the practitioner who values the utility of collective and emergent behaviors in applications as broad as mass sensing, signal processing, pattern generation, and even neural computing.

3.1 A Discrete System of Coupled Resonators

The system under consideration consists of an electromagnetically transduced microresonator array, actuated by the Lorentz force, due to the interaction between an

39 external permanent magnet and the current loop which is integrated on the resonator [1, 147]. Global dissipative coupling arises naturally due to the current that identically flows through each individual resonator and the resulting electromagnetic interactions. In this work reactive coupling is also included in the model for the sake of generality. Experimental results indicate that the parametrically excited systems exhibit moderately large amplitudes near resonance and modeling must allow large elastic displacements, making it a good case to include cubic nonlinearity into the model [5, 148].

3.1.1 Equations of Motion

Here the physical model as used by authors in [1, 147] have been considered for understanding the dynamics of coupled resonators. It is reasonable to assume that first mode of cantilever is dominant and is prevalent to do so [5,31], using thelumped mass approximation the governing equation of discrete coupled resonator system is written as,

α˜ ∑N β˜ ∑N m z¨ + c z˙ + k z + k z3 − z˙ − z = f˜(t), (3.1) i i i i 1,i i 3,i i N j N j i j=1 j=1 where each individual resonator in the N-element array is characterized by the index i and has an associated displacement zi. Likewise, mi, ci, and k1,i represent the mass, damping, and linear stiffness of each resonator, while k3,i is the coefficient of the cubic nonlinearity. The parameters α˜ and β˜ characterize the magnitude of the dissipative and reactive global coupling respectively, and finally each resonator is subject toan

˜ external time-dependent excitation represented by fi(t). 40 3.2 Continuum Model

The attempt is made in order to reduce the coupled differential equations in the previous section to an integro-differential equation which can represent both finite and infinite models. The obtained equation would take care of all the possible parameter variations which include that of coupling, can be solved by applying standard analytical techniques of non-linear dynamics.

3.2.1 Formulation

The discrete system of N coupled resonators in Eq. (3.1) can be written in a continuum formulation, leading to a single integro-differential equation to represent dynamics of the population of resonators [142, 149]. The index i used to represent a resonator in the discrete setup can be replaced by an indexing variable si so that zi(t) ≡ z(si; t), while mi ≡ m(si), k1,i ≡ k1(si), etc., and without loss of generality the index s ∈ [0, 1]. In addition the global reactive and dissipative coupling terms in

Eq. (3.1) can be expressed as integrals over the population of resonators as ∫ ( ) α˜ ∑N 1 ∑N δ(n − s ) z˙ =α ˜ z˙(n; t) j dn, (3.2) N j N j=1 0 j=1 where δ(x) is the Dirac delta function. The summation within this integrand can be identified as ρN (s), representing the contribution of individual resonators to the global coupling, so that

1 ∑N ρ (s) ≡ δ(s − s ). (3.3) N N j j=1

41 Therefore, the discrete system given in Eq. (3.1) can be expressed as a single integro- differential equation of the form

3 ˜ m(s)z ¨(s; t) + c(s)z ˙(s; t) + k1(s) z(s; t) + k3(s) z (s; t) = f(s; t) ∫ ∫ 1 1 ˜ +α ˜ z˙(n; t) ρN (n) dn + β z(n; t) ρN (n) dn, (3.4) 0 0 with s = si, i = 1,...,N. Note that even for a discrete system of resonators this representation is valid for any s, although with the discrete population density defined in Eq. (3.3) only those resonators with s = si contribute to the global coupling.

Although resonators with index s ≠ si are still defined, they do not contribute to the global coupling function but are driven by the population of resonators in a one- way fashion. This discrete formulation can be extended to a generalized continuum description, where the specific distribution ρN (s) present in Eq. (3.4) is replaced by a general population density function ρ(s), with ∫ 1 ρ(n) dn = 1. (3.5) 0

This system can be scaled in terms of the mean mass and stiffness, identified as ∫ ∫ 1 1 ⋆ ⋆ m = m(s) ds, k = k1(s) ds, (3.6) 0 0 so that

⋆ ⋆ m(s) = m (1 + µ(s)), k1(s) = k (1 + σ(s)). (3.7)

Here µ(s) and σ(s) represent the distribution of population mass and stiffness with respect to the corresponding mean values. Eq. (3.4) is then written in terms of new 42 variables as

√ (1 + µ(s))z ¨(s; t) + 2 κ ζ(s)z ˙(s; t) + κ (1 + σ(s)) z(s; t) + γ(s) z3(s; t) ∫ ∫ 1 1 = f(s; t) + α z˙(n; t) ρ(n) dn + β z(n; t) ρ(n) dn, (3.8) 0 0 where k⋆ c(s) k (s) f˜(s; t) κ = , ζ(s) = √ , γ(s) = 3 , f(s, t) = , m⋆ 2 κ m⋆ m⋆ m⋆ (3.9) α˜ β˜ α = , β = . m⋆ m⋆ For physically realistic parameter values the damping is weak, indicated with the introduction of a small scaling coefficient ϵ, so that ζ(s) → ϵ ζ(s). Likewise the mass and stiffness distributions, nonlinearity and coupling are assumed tobe O(ϵ), so that µ(s) → ϵ µ(s), etc. Finally, the array of resonators is assumed to be subject to a uniform weak external magnetic field. Each resonator in the system is excited harmonically at frequency Ω and amplitude ϵ F0 so that

f(s, t) = ϵ F0 sin Ωt. (3.10)

The non-dimensional stiffness κ can be related to frequency of excitation Ω using frequency detuning ω by writing it as

κ = (Ω + ϵ ω)2. (3.11)

Finally the scaled equation can be written to O(ϵ) as

(1 + ϵ µ(s))z ¨(s; t) + 2ϵΩ ζ(s)z ˙(s; t) ( ) + Ω2 + ϵΩ (Ω σ(s) + 2 ω) z(s; t) + ϵγ(s) z3(s; t) [ ∫ ∫ ] 1 1 = ϵ F0 sin Ωt + α z˙(n; t) ρ(n) dn + β z(n; t) ρ(n) dn . (3.12) 0 0 43 3.2.2 Perturbation solution

Eq. (3.12) represents the dynamics of a resonator with an index s, described as a

Duffing-like system, under the influence of global coupling and external excitation.

The method of multiple time scales is used to analyze this system by identifying time scales τ = Ωt (fast time) and η = ϵt (slow time), so that

d ∂ ∂ = Ω + ϵ , dt ∂τ ∂η (3.13) d2 ∂2 ∂2 = Ω2 + 2ϵΩ + O(ϵ2), dt2 ∂τ 2 ∂τ ∂η while the response of each resonator is expanded as a series in ϵ as

2 z(s; τ, η) = z0(s; τ, η) + ϵ z1(s; τ, η) + O(ϵ ). (3.14)

Substitution of this expansion and the multiple time scales into Eq. (3.12), and col- lecting terms with coefficients of ϵiyields

O(ϵ0):

∂2z 0 + z = 0, (3.15a) ∂τ 2 0

O(ϵ1):

∂2z 1 [ [ ] 1 − 2 − 3 2 + z1 = 2 F0 sin τ Ω σ(s) + 2ω Ω z0 γ(s) z0 ∂τ Ω ∫ ∫ 1 ∂z (n; τ, η) 1 + α Ω 0 ρ(n) dn + β z (n; τ, η) ρ(n) dn ∂ τ 0 0 0 ] ∂z ∂2z ∂2z −2ζ(s)Ω2 0 − 2Ω 0 − Ω2 µ(s) 0 . (3.15b) ∂τ ∂η ∂τ ∂τ 2

To lowest order the response of z(s; τ, η) is

z0(s; τ, η) = A(s; η) sin(τ) + B(s; η) cos(τ), (3.16)

44 where (A(s; η),B(s; η)) are yet to be determined but independent of the fast time scale τ. Introducing z0 into next order of the approximation leads to secular terms in the system—for uniformly valid solutions these must be eliminated which provide equations on the slow time scale η that determine the evolution of (A(s; η),B(s; η)).

These slow flow equations can be written as

∂A 1 [ = −2ζ(s)Ω2 A + Ω [(µ(s) − σ(s)) Ω − 2ω] B ∂η 2Ω ∫ ∫ 1 1 + α Ω A(n) ρ(n) dn + β B(n) ρ(n) dn 0 0 ] 3γ(s) [ ] − B A2 + B2 , (3.17a) 4

∂B 1 [ = − Ω [(µ(s) − σ(s)) Ω − 2ω] A + 2ζ(s)Ω2 B ∂η 2Ω ∫ ∫ 1 1 + F0 − α Ω B(n) ρ(n) dn + β A(n) ρ(n) dn 0 0 ] 3γ(s) [ ] − A A2 + B2 . (3.17b) 4

3.2.3 Stationary solutions

The steady-state behavior of the population can be obtained by setting differential terms in Eqs. (3.17) to 0, so that A(s; η) → A0(s) and B(s; η) → B0(s). The integral terms can be represented as ∫ ∫ 1 1 V ≡ A0(n) ρ(n) dn, W ≡ B0(n) ρ(n) dn, (3.18) 0 0

2 ≡ 2 2 while R (s) A0(s) + B0 (s), which represents the steady-state amplitude of the resonator with index s. Finally, these equilibrium equations can be explicitly written 45 as

2 (2ζ(s)Ω ) A0(s) − X(s) B0(s) = G1, (3.19) 2 X(s) A0(s) + (2ζ(s)Ω ) B0(s) = G2, with

G1 ≡ α Ω V + β W, G2 ≡ −F0 + α Ω W − β V, (3.20) 3γ X(s) ≡ Ω [(µ(s) − σ(s)) Ω − 2ω] − R2(s). 4

Note that G1 and G2 include the excitation and global coupling, while X(s) describes the influence of the mistuning and nonlinearity on the stead-state response. These equations can be combined so that

G2 + G2 R2(s) = 1 2 , (3.21) X2(s) + (2ζ(s)Ω2)2 with

2ζ(s)Ω2 G + X(s) G A (s) = 1 2 R2(s), 0 G2 + G2 1 2 (3.22) 2 − 2ζ(s)Ω G2 X(s) G1 2 B0(s) = 2 2 R (s). G1 + G2 Note that in general X(s) depends on the amplitude R(s) through the nonlinearity while G1 and G2 are functions of the global coupling and depend on A(s) and B(s).

Therefore the equilibrium distribution is only implicitly defined by this expression.

Note however that this expression is identical to that for an uncoupled resonator subject to forcing amplitude G⋆, with

2 ≡ 2 2 G⋆ G1 + G2, (3.23) ( )( ) ≡ 2 − − 2 2 2 2 F0 2 (αΩ W β V ) F0 + (αΩ) + β V + W .

46 Therefore G⋆ represents the effective forcing amplitude on each resonator in the population, so that all resonators are identically forced and will be called as global coupling force from now.

For a given population of resonators and coupling parameters, the steady- state distribution is determined from Eqs. (3.18, 3.21 and 3.22) using MATLAB. The optimization function fmincon is used to minimize the objective function [ ∫ ] [ ∫ ] 1 2 1 2 E(V,W ) = V − A0(n) ρ(n) dn + W − B0(n) ρ(n) dn , (3.24) 0 0 where given test values for (V,W ), the amplitude function R(s) and therefore A0(s) and B0(s) are determined for each index s in the population. With nonzero values of γ, this system can exhibit multiple solutions for a given set of system parameters, as individual resonators can be driven to bi-stability with increasing values of G⋆.

However, since G⋆ depends on the population through V and W each equilibrium distribution must be determined uniquely.

3.2.4 Stability Analysis

In order to determine the stability about equilibrium solution for coupled resonators

[138], the slow flow amplitude components can be written as,

A(s; η) = A0(s) + u(s; η),B(s; η) = B0(s) + v(s; η), (3.25)

47 and linearizing the Eq. (3.17) in u and v as, [ ( ) ∂u 1 3γ(s) = − 2ζ(s)Ω2 + A (s) B (s) u(s; η) ∂η 2Ω 2 0 0 ( ) 3γ(s) [ ] + Ω [(µ(s) − σ(s))Ω − 2ω] − A2(s) + 3 B2(s) v(s; η) 4 0 0 ∫ ∫ ] 1 1 +α Ω u(n; η) ρ(n) dn + β v(n; η) ρ(n) dn , (3.26a) 0 0

[ ( ) ∂v 1 3γ(s) [ ] = − Ω [(µ(s) − σ(s))Ω − 2ω] − A2(s) + 3 B2(s) u(s; η) ∂η 2Ω 4 0 0 ( ) 3γ(s) − 2ζ(s)Ω2 + A (s) B (s) v(s; η) 2 0 0 ∫ ∫ ] 1 1 −β u(n; η) ρ(n) dn + α Ω v(n; η) ρ(n) dn . (3.26b) 0 0

Solutions to Eq. (3.26) can be written as     u(s; η) p(s)     λ η   =   e , (3.27) v(s; η) q(s) where the eigenvalue λ is independent of the population index s, by substituting

Eq. (3.27) into Eq. (3.26) leads to the integral characteristic equation

(λI − J(s)) ψ(s) = K Ψ, (3.28)

∫ 1 with Ψ = 0 ψ(n) ρ(n) dn, where 3γ(s) [J] = 2ζ(s)Ω2 + A (s) B (s), 11 2 0 0 3γ(s) [ ] − − − 2 2 [J]12 = Ω [(µ(s) σ(s))Ω 2ω] + A0(s) + 3 B0 (s) , 4 (3.29) 3γ(s) [ ] [J] = Ω [(µ(s) − σ(s))Ω − 2ω] + 3 A2(s) + B2(s) , 21 4 0 0 3γ(s) [J] = 2ζ(s)Ω2 − A (s) B (s), 22 2 0 0

48 and   α Ω β    K =   . (3.30) −β α Ω

In terms of the integral terms Ψ, this characteristic equation for λ can be written as [ ∫ ] 1 − I − (λ I − J(n)) 1 ρ(n) dn · K Ψ = 0. (3.31) 0

For non-trivial solutions of ψ(s), the value λ in Eq. 3.28 is found by equating det(λ I−

J(s)) = 0, this is independent of coupling parameters K exist and this λ is generally called as continuous spectrum. In contrast the discrete spectra satisfies (∫ ) 1 [ − ] det I − (λ I − J(n)) 1 · K ρ(n) dn = 0. (3.32) 0

Therefore the discrete spectrum depends explicitly on the global coupling through K.

As continuous spectrum amount to stability of individual uncoupled resonators, discrete spectrum represents to collective stability of the whole array at that equilibrium state.

3.3 Results

1 2 2 B0(s) The amplitude response R(s) = (A (s) + B (s)) 2 and phase tan Φ(s) = can 0 0 A0(s) be obtained from Eq. (3.22) for the values (V,W ) which satisfy Eq. (3.24). It was previously mentioned that s is a continuous variable in the interval [0, 1], and for a discrete system s ≡ si, where i = 1 , 2 ,...N. A uniform distribution of the

49 8 4

6 2

4 (s) 0 R(s)

2 -2

0 -4 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

(a) R(s) vs m(s) (b) Φ(s) vs m(s)

Figure 3.1: Response for a mono-stable solution γ = 0.02, bc bc , for coupling parameters (α , β) = (0.5 , 1.0).

population, index and density function can be respectively defined as,

i − 1 1 ∑N s = , ρ(s) = δ(s − s ). (3.33) i N − 1 N i i=1

Subsequently the parameters used for simulations N = 10 resonators are as chosen as,

Ω = 1,F0 = 2.00, ω = 0, ϵ = 0.01, (3.34)

ζ(si) = 0.25, σ(si) = 0, µ(si) = −10 + 20 si(mass detuning).

When γ = 0.02 the only equilibrium state that exists is shown in Figure 3.1, here all the resonators in the population have X(s) (describing the influence of mistuning and non-linearity) which satisfy the Eq. (3.22) and Eq. (3.24) at coupling values of

(α , β) = (0.5 , 1.0).

50 8

4 R(s)

0 0.9 0.95 1 1.05 1.1 m(s)

Figure 3.2: R(s) vs m(s) for global coupling constants (V,W ) = (0.3840 , −0.3298), given parameters γ = 0.2 , (α , β) = (0.5 , 1.0) X(s) > 0, X(s) > 0(unstable branch), X(s) = 0, X(s) < 0

When the Duffing response exhibits multiple states for a higher valueof γ, the variable X(s) can have multiple states in Eq. (3.20). Once the global-self consistency variables (V,W ) are determined, the response R(s) of continuous system is shown in Figure 3.2. The response can be separated into branches of X(s) < 0 and X(s) >

0, X(s) > 0 is further divided in to stable and unstable parts indicated by solid and dashed lines respectively. The dotted curve between these branches also called backbone curve, corresponds to solving Eq. (3.21) for R(s) with X(s) = 0.

For a different set of (V,W ), there can exists another unique response curve of X(s), even in the case of discrete resonators, each of the ith resonator in the population can have a different signs for X(si). Each such set of X(si) can be used to evaluate the corresponding R(si) and Φ(si) (A(si) ,B(si)) making an equilibrium distribution. In order to indicate such distributions, the following naming scheme

51 is proposed: the first resonator on X(s) > 0 branch among the population of resonators is used to indicate the equilibrium distribution and its location on the continuous curve X(s) is used to denote an equilibrium distribution, suppose resonators si . . . sN have X(si) > 0 and the resonators s1 . . . si−1 have X(si) < 0 then the distribution would be called as Ei, where subscript i represents the first resonator on

X(s) > 0 branch and if the resonator si happens to be on the unstable part of the

E u curve a superscript u is used and indicated as i . This can be demonstrated with an example, at γ = 0.08 three equilibrium distributions can be seen in Figure 3.3.

The distribution shown using bc bc has resonator s6, . . . s10 have X(s) > 0 and the resonators s1, . . . s5 have X(s) < 0, since the first resonator on X(s) > 0 is s6, hence it is named as E6. Similarly the response shown using bc bc has resonator s7, . . . s10 have X(s) > 0 and the resonators s1, . . . s6 have X(s) < 0, since the first resonator on X(s) > 0 is s7, hence it is named as E7. The one indicated by bc bc has resonator

E U i = 6 on the unstable part of X(s) > 0 hence it is denoted by 6 . This nomenclature has been followed for all the equilibrium distributions discussed in the rest of this dissertation.

The spectrum as defined from Eq. (3.32) for a finite number of resonators is obtained by solving for λ, the eigenvalues which are the roots of Eq. (3.32) cannot be evaluated in the closed-form, they are dependent on the initial guess on that is supplied to fsolve (equation solver in MATLAB). For the only equilibrium state at

γ = 0.02, the spectrum plot of eigenvalues are shown in Figure 3.4, among these an eigenvalue of Re(λ) = −0.2141 is the closest value to zero. Though these are obtained

52 8 4

6 2

4 (s) 0 R(s)

2 -2

0 -4 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

(a) R(s) vs m(s) (b) Φ(s) vs m(s)

4 R(s)

0 0.98 1 1.02 1.04 m(s)

bc bc E bc bc E U bc bc E Figure 3.3: Multiple solutions at γ = 0.08, 6, 6 , 7, for coupling parameters (α , β) = (0.5 , 1.0) .

53 10

5 ) 0 Im(

-5

-10 -0.8 -0.6 -0.4 -0.2 0 0.2 Re( )

Figure 3.4: Re(λ) vs Im(λ), γ = 0.02 (α , β) = (0.5 , 1.0) bc bc Discrete spectrum.

by solving for λ using Eq. (3.32) but these are more close to continuous spectrum of

Eq. (3.28) indicating these are not influenced by coupling.

Similarly when γ = 0.08, there are three equilibrium states as discussed previously, for which the discrete spectrum plots can be seen in the Figure 3.5. For the equilibrium distribution E6 discrete spectrum plot Figure 3.5a, all the eigenvalues have Re(λ) < 0 are near to −0.22, though the imaginary part varies along [−6 , 6], hence this equilibrium state is stable. The discrete spectrum plot in Figure 3.5b is

E U for the equilibrium distribution 6 , two eigenvalues among them have zero real part and one of them is positive, this clearly indicates that this configuration is unstable.

In Figure 3.5c the discrete spectrum plot for E7 is shown, this clearly indicate that this equilibrium configuration is stable since the real part of all the eigenvalues isless than zero, the imaginary part of these eigenvalues vary in the range of [−6 , 6].

54 10 10

5 5 ) ) 0 0 Im( Im( -5 -5

-10 -10 -0.8 -0.6 -0.4 -0.2 0 0.2 -0.8 -0.6 -0.4 -0.2 0 0.2 Re( ) Re( )

(a) bc bc Discrete Spectrum (b) bc bc Discrete Spectrum

5 ) 0 Im( -5

-10 -0.8 -0.6 -0.4 -0.2 0 0.2 Re( )

E E U E E Figure 3.5: Spectrum plot Re(λ) vs Im(λ), γ = 0.08 a) 6, b) 6 and c) 7 , both 6

E E U and 7 have eigenvalues with Re(λ) < 0; 6 has both the spectrums with Re(λ) > 0

55 CHAPTER IV

MONOSTABLE DISTRIBUTIONS

4.1 Introduction

In the previous chapter, the analytical modeling of Large Array resonators has been discussed, this includes the continuum method for formulating a discrete and continuous systems, using the method of multiple scales the scaled governing equations are analyzed. The slow flow equations involving the amplitude components are obtained from multiple scales, the procedure for obtaining multiple equilibrium states has been discussed along with the stability criteria. It has to be noted that the non-linear integral equations may have multiple equilibrium states, depending on the non-linear stiffness coefficient γ and coupling parameters. That happens only when γ is sufficiently large, discussion on multiple equilibrium states and their stability is done in succeeding chapter, but when γ ≈ 0 or relatively small, the solution is always unique, analysis of such mono-stable states is discussed here. The influence of dissipative and reactive coupling on the amplitude and phase of the array are also discussed here.

Global self-consistency variables V,W which influence the response is also analyzed here, the effect of population size N on the dynamics of the system has also been studied.

56 4.2 Results

Eqs. (3.21) and (3.22) can be reduced to a cubic non-linear equation in X(s), hence for a discrete system of N resonators, there can be N 3 values of X(s). Only the real roots of X(s) would lead to multiple equilibrium states represented by A(s) ,B(s), but when γ is small or close to zero the roots are mostly imaginary. Then only one set of X(si), which satisfies the Eq 3.22 can be obtained leading to equilibrium states that are unique, these are called monostable solutions.

The analytical predictions for the stationary solutions can be compared against results obtained from direct numerical simulation of the discrete equations of motion,

Unless noted, here we choose ϵ = 10−2; the remaining default parameters for the system are

m0 = 1, Ω = 1, c = 0.50,F = 1, (4.1) α = 2.00, β = 5.00.

With γ = 0 the system is linear and the resulting equations of motion admit an exact solution. [147] developed the solution for a discrete system of resonators, while for a general population, the solution to Eqs. (3.12) can be solved in closed form. In particular, for a discrete system of resonators with distribution parameters si, i =

1,...,N, the global self-consistency coupling variables reduce to ∫ ∫ ∞ 1 ∑N ∞ 1 ∑N V = A(n) ρ(n) dn = A(si),W = B(n) ρ(n) dn = B(si), −∞ N −∞ N i=1 i=1 (4.2)

57 In what follows the response of the system is illustrated for a discrete system of resonators, as both the system parameters and number of. With a discrete system as given above, defined by the values si, the integral equations for [A(s),B(s)] can be solved for any value of s, but only those resonators with s = si influence the global self-consistency coupling variables V,W in Eq. (4.2).

4.2.1 Uniform Distribution, N = 100

For illustrative purposes, a uniform distribution of N = 100 discrete resonators represented by Eq. (3.3) is chosen with si equally spaced within the interval ϵ s ∈

[−0.20, 0.20]. The amplitude and phase response functions, R(s) and Φ(s), are shown in Figure 4.1 as a function of the resonator mass m = m0(1 + ϵ s) with the default parameters of the system. The results of iterative solution to the amplitude equations given in Eqs. (3.22) are represented by the solid line, while the stationary amplitudes obtained from direct numerical simulation of the discrete equations of motion are shown with the marked points.

The kernels of the global self-consistency coupling constants (V,W ), that is, A(s) ρ(s) and B(s) ρ(s), are shown in Figure 4.2 as obtained from the iterative solution. Note that with a uniform distribution these simply reduce to A(s) and

B(s), scaled by the number of oscillators. Consequently, for this distribution, the resonators near s = 0 (m = m0) contribute most significantly to the global self- consistency coupling variables.

In Figure 4.3 and Figure 4.4 the amplitude and phase response functions

58 1 π 10 2 (a) (b) bC bC bC bC 0 bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC 0 bC bC bC bC bC bC bC 10 bC bC bC bC bC bC bC ) bC bC ) bC bC

s bC bC s π bC bC bC bC − bC

( bC bC bC bC bC bC bC bC bC bC 2 bC bC bC bC bC bC

bC bC Φ( R bC bC bC bC bC bC bC bC bC − bC bC bC bC bC bC bC bC 1 bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC 10 bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC − bC bC bC bC π bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC

−2 − 3 π 10 2 0.8 0.9 1.0 1.1 1.2 0.8 0.9 1.0 1.1 1.2 m(s) m(s)

Figure 4.1: Linear response functions (α = 2.00, β = 5.00, γ = 0, N = 100); iterative solution, ⊗ ⊗ ⊗ direct numerical simulation; (a) R(s), (b) Φ(s).

× −2 × −2 1 10 1 10 b b b b b (a) (b) b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b 0 b b b b 0 b b b b b b b b b b b b b b b −1 b −1 b b A ρ B ρ b b

−2 −2 b

−3 −3 0.80 0.90 1.00 1.10 1.20 0.80 1.1 1.2 1.3 1.4 m m

Figure 4.2: Coupling kernels (α = 2.00, β = 5.00, γ = 0, N = 100, iterative solution);

(a) A(s) ρ(s), (b) B(s) ρ(s).

59 π 3.0 2 (a) (b) 0 2.0 ) ) s s − π ( 2 Φ( R 1.0 −π

− 3 π 0 2 0.95 1.00 1.05 0.95 1.00 1.05 m(s) m(s)

Figure 4.3: Linear response functions varying α (β = 10.00, γ = 0, N = 100);

α = 0 , α = 4.0 , α = 8.0; (a)R(s), (b)Φ(s)

are shown for various combinations of α and β. Specifically,in Fig. 4.3 the response functions are shown for varying with β = 10.00; in Figure 4.4 the response functions are shown for varying with α = 4.00. As these coupling parameters change the amplitude response functions are primarily scaled by the maximum response. In contrast, for different values of α and β the phase response function is uniformly shifted for all values of m, indicating that the global coupling introduces an additional phase shift of the resonator population relative to the excitation. With β = 0 global coupling is purely dissipative, and the maximum amplitude is shown in Figure 4.5a as the dissipative coupling parameter α is varied. In contrast, α = 0 corresponds to purely reactive coupling, and the maximum amplitude is shown in Figure 4.5 as the reactive coupling parameter β is varied. The addition of such reactive coupling reduces the maximum amplitude of the response.

Similarly the global coupling force which represents G∗ represents the effective forcing amplitude on each resonator in the population, so that all resonators are

60 π 3.0 2 (a) (b) 0 2.0 ) ) s s − π ( 2 Φ( R 1.0 −π

− 3 π 0 2 0.95 1.00 1.05 0.95 1.00 1.05 m(s) m(s)

Figure 4.4: Linear response functions varying β (α = 4.00, γ = 0, N = 100); β =

0 , , β = 10.0 , β = 20.0; (a)R(s), (b)Φ(s)

25 bC 2.5 (a) (b) 20 2.0 bC bC bC bC bC bC bC bC bC 15 bC 1.5 bC bC bC max bC max bC bC bC bC bC R 10 R 1.0 bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC 5 bC bC bC 0.5 bC bC bC bC bC bC bC bC bC bC bC bC bC 0 0 0 2 4 6 8 10 12 0 20 40 α β

Figure 4.5: Maximum response amplitude Rmax, varying coupling parameters (γ = 0,

N = 100); iterative solution, bC bC bC direct numerical simulation; (a) dissipative coupling (β = 0), (b) reactive coupling (α = 0).

61 1.5 (a) (b) 10 bC bC bC bC bC bC 1.0 bC bC bC bC * ∗ bC bC bC bC G G bC bC bC bC 5 bC bC bC bC bC bC bC 0.5 bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC bC 0 bC bC bC bC bC bC bC bC bC bC bC 0 0 2 4 6 8 10 12 0 10 20 30 40 50 α β

Figure 4.6: Global coupling force G∗ acting on all the resonators in the array, varying coupling parameters (γ = 0, N = 100); iterative solution, bC bC bC direct numerical simulation; (a) dissipative coupling (β = 0), (b) reactive coupling (α = 0).

identically forced is shown in Figure 4.6, as deduced previously with increase in dissipative coupling at β = 0 increases the global coupling force acting on array of resonators as seen in Figure 4.6a similarly increasing in reactive coupling at α = 0 decreases the global coupling force G∗ that is shown in Figure 4.6b.

While the linear system admits an exact solution, for γ ≠ 0 no such closed- form expression exists. However, the amplitude and phase response functions can still be determined with the iterative solution. In Figure 4.7 the phase response functions for the uniform distribution of resonators are shown for three different values of γ. As expected, γ < 0 corresponds to a softening response, so that near the resonant frequency the amplitude function is shifted to the left relative to the linear system, while γ > 0 corresponds to a hardening system. As γ increases the frequency- amplitude shifts become more significant, and can lead to bi-stability in the system response. Furthermore, as γ varies the maximum amplitude of the response varies as

62 π 3 2 (a) (b) rs 0 ut ut ut ut ut ut ut bcrs bcrs bcrs bcrs rsbc rsbc rsbc rsbcut rsbcut bcrsut bcrsut bcrsut bcrsut bcrsut rsbcut rsbcut rsbcut rsbcut bcrsut bcrsut bcrsut rsbcut rsbcut bcrsut bcrsut rsbcut rsbcut bcrsut bcrsut rsbcut bcrsut rsbcut bc bcrsut bc rsbcut rs bcrsut bcrsut rsbcut rsbcut rsbcut bcrsut ut ut rsbc rsbcut bcrsut bcrsut bcrsut bcrsut bcutrs 2 bcrs rs ut bcrs ) ) ut rs ut s ut s π − bc rs ( rsbc bc 2 ut ut Φ( R rs bc rs utbcrs bc ut ut bc 1 rs ut rs utbcrs bc bc bcutrs ut ut − bcrsut bcrsut rs rs π rsbcut rsbcut bcrsut bcrsut bcut bc bc rs bcrsut bcrsut bcrsut rsbcut rsbcut bcrsut rsbcut bcrsut rsbcut bcrsut bcrsut rsbcut bcrsut bcrsut bcut bcut ut ut ut ut ut ut rs rs rs bcrs bcrs bcrs rsbc rsbcut rsbcut bcrsut bcrsut bcrsut rsbcut rsbcut rsbcut rsbcut rsbcut bcrsut bcrsut bcrsut bcrsut bcrsut bcrsut rsbcut rsbcut rsbcut bcrs bc bcrs ut ut bcrs bcrs ut ut bcrs rs bcrs ut ut bcrs rs rs bcrs utbc ut ut utbc bcrs rs rs rs utbcrs utbc ut ut utbc utbcrs rs rs rs rs bcrs utbcrs utbcrs utbc utbc utbc utbc utbcrs utbcrs bcrs rs rs rs rs bcrs bcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbc utbc ut ut utbc utbc utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs bcrs bcrs rs rs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbc utbc ut ut − 3 π ut ut utbc utbc utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs utbcrs 0 2 0.98 0.99 1.00 1.01 1.02 0.98 0.99 1.00 1.01 1.02 m(s) m(s)

Figure 4.7: Nonlinear response functions, varying γ (α = 2.00, β = 5.00, N = 100,

iterative solutions); ut ut γ = −0.10, bc bc γ = 0, rs rs γ = 0.10; (a) R(s), (b) Φ(s).

well. This is in contrast to the response of a single resonator with cubic nonlinearity,

in which the location in m at which the maximum amplitude occurs shifts, but the

value of Rmax remains constant.

4.2.2 Other Distributions

Perhaps most importantly, the formulation and analytical framework presented herein

allow for the consideration of arbitrary distributions of resonators through the pop-

ulation density function ρ(s). As discussed above, the global coupling influences the

individual resonators only through the values of V and W , as determined by the

self-consistency condition given in Eq. (4.2). Hence, the iterative approach described

above can be applied for any arbitrary resonator distribution.

The response of the system can be considered as the number of discrete

resonators varies. In Figure 4.8 the response functions are shown for two uniform

distributions, one with N = 20 and the second with N = 1000. We note that the

63 1 π 10 2 (a) (b)

0 bC bC bC bC bC bC bC bC bC 0 bC 10 bC bC ) ) s bC bC s − π ( bC bC 2

bC bC Φ( R bC bC −1 bC bC 10 bC bC bC bC bC bC bC bC bC −π bC bC bC bC bC bC bC bC bC

−2 − 3 π 10 2 0.8 0.9 1.0 1.1 1.2 0.8 0.9 1.0 1.1 1.2 m(s) m(s)

Figure 4.8: Response functions (α = 2.00, β = 5.00, γ = 0.05, iterative solutions);

N = 1000, bC bC bC N = 20; (a) R(s), (b) Φ(s).

response of any individual resonator is similar for these two distributions, described by Eq. (3.19) depends only on the values of V,W defined by Eq. (3.18).

As shown in Figure 4.9, as N varies the values of the global self-consistency coupling variables V and W vary significantly for small N, before converging as N increases. Thus, above approximately N = 50 the discrete population behaves as a continuum distribution. As N increases (V,W ) → (−0.0286, −0.0824).

For the discrete system, the population distribution is defined by the values of si defined in Eq. (3.3). A random distribution of resonators is determined bychoosing si over the interval si ∈ [0.80, 1.20] with equal probability. Each different distribution results in different values of (V,W ), but recall that the values of these global self-consistency coupling variables are determined from the self-consistency condition given in Eq. ( 3.18). In Figure 4.10, 500 different realizations of the aforementioned distribution are shown for N = 20 and N = 500 discrete resonators. The spread in

(V,W ) is more significant for N = 20 as compared to N = 500 indicating that the

64 0.1 0.1

(a) b (b)

b b b b b b b b b b b b b b 0 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b 0 b b b b b b b b b b b b b b b b b − b b b b b V 0.1 b b b b W b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b − b b b b b b 0.1 −0.2 b b

−0.3 −0.2 0 20 40 60 0 20 40 60 N N

Figure 4.9: Global self-consistency coupling variables, varying N (α = 2.00, β = 5.00,

γ = 0, iterative solutions); (a) V , (b) W .

stationary response is more sensitive to the population distribution for fewer numbers of resonators. For reference the values obtained for the corresponding uniform continuum distribution of resonators are also marked in each figure.

Finally, for a normal distribution of resonators, the global self-consistency coupling variables are shown in Figure 4.11 as the standard deviation of the distribution σ varies. Here, the resonator parameters si of the discrete distribution are chosen to equally partition the cumulative normal distribution function. Note that in the limit σ → 0, each resonator in the population becomes identical and

(V,W ) → (−0.1835, 0.0550).

The stability of monostable distributions is determined from discrete spectrum as mentioned in Eq. (3.31), where the sign of real part of eigenvalues need to be checked for stability. On a plane of coupling parameters the stable and unstable region in case of a uniform distribution is as shown in Figure 4.12, the region to left of this curve is stable while the region to right of the curve is unstable. It can be

65 0.50 0.50 (a) (b)

0.25 0.25

bc bc bc bc bc bc bcbc bcbc bc bc bcbcbc bcbc bcbcbc bc bc bc bcbc bc bc bcbcbc bcbcbcbc bcbc bcbc bcbcbcbc bcbc bc bc bcbc bc bc bc bcbc bc bcbc bc bc bcbcbc bc bc bc bc bc bc bcbcbcbc bcbc bc bc bcbc bc bcbcbc bc bcbcbc bc bc bcbcbcbcbcbcbcbcbcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bcbc bcbcbcbcbc bc bc bcbcbcbcbc bc bc bcbc bc bcbc bc bc bc bcbcbc bcbcbc bcbcbc bc bc bc bc bc bc bc bc bcbc bc bc bc 0 bc bc bc bc bc bc bcbc bcbc bc bcbc bc bc bcbcbc bc bc bc bc bc bcbc bcbc bc bc bcbc bc bc bc bc bc bc bc bc bc bc 0 bc bc bc bc bc bc bc bc bc bc bc bc W bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bcbc bcbc bc bc bcbc bc bc bc bc bc bc bc W bc bc bcbc bc bcbcbc bcbcbc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bcbcbcbc bcbc bcbc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbc bcbcbc bcbcbcbcbcbc bcbcbc bcbcbcbc bc bc bc bc bc bcbc bc bc bc bcbc bc bc bc bc bcbc bc bc bc bcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbc bcbc bc bc bc bc bc bc bc bc bcbcbcbcbc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bcbc bcbcbcbc bcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbc bc bcbcbc bc bc bc bc bc bc bc bcbc bc bc bcbcbc bc bc bc bc bc bcbcbc bcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbc bcbc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbcbc bc bcbcbcbcbcbc bcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbc bcbcbcbcbcbcbcbc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bcbc bc bcbcbcbcbcbc bcbcbc bc bcbcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbc bc bc bc bc bc bcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc − bc − 0.25 bc 0.25 bc bc bc

bc bc bc bc

−0.50 bcbc −0.50 bc −0.50 −0.bc 25 0 0.25 0.50 −0.50 −0.25 0 0.25 0.50 bc bc V V

bc Figure 4.10: Global self-consistency coupling variables (α = 2.00, β = 5.00, γ = 0,

iterative solutions, 500 random distributions); (a) N = 20, (b) N = 500. The large

point indicates the values obtained from the uniform distribution of resonators.

0.1 0.1 (a) (b) 0 0

− −

V 0.1 0.1 W

−0.2 −0.2

−0.3 −0.3 0 0.05 0.10 0.15 0.20 0 0.05 0.10 0.15 0.20 σ σ

Figure 4.11: Global self-consistency coupling variables, iterative solutions, normal

distribution varying σ (α = 2.00, β = 5.00, γ = 0, N = 500); (a) V , (b) W .

66 40

β 20

0 0 2 4 6 8 10 12 14 α

Figure 4.12: A curve separating the stable(left) and unstable(right) regions on an

α , β plane, for an uniform distribution, γ = 0, N = 100.

noticed that for lower values of α the value of β at which a distribution losses stability is higher and for values α below a certain point, the region is always stable, since increasing β reduces the amplitude of distribution as witnessed previously.

4.3 Conclusions

This chapter presents the analysis of large degree-of-freedom, coupled dynamical systems are monostable resonators here, wherein the asymptotic state is unique, the methodology discussed in previous chapters in the analysis of a wide variety of complex coupled dynamical systems. Increasing reactive coupling decreases the amplitude response and G∗, while increasing dissipative coupling leads to increase in amplitude response. The influence of self-consistency variables (V,W ) on coupling in case of different distributions has been thoroughly discussed.

67 CHAPTER V

EFFECT OF COUPLING PARAMETERS ON MULTIPLE EQUILIBRIUM

STATES

The nonlinearities present in the individual resonators can lead to multiple coexisting equilibrium distributions, particularly as the parameters associated with the global coupling vary. This chapter describes the mechanisms that give rise to multiple equilibrium states as well as the affect of global coupling on the resulting stability of these equilibrium states as well as the global self-consistency variables.

In what follows a discrete population of N = 10 resonators is considered, and unless noted each resonator is identical except for the individual masses, with

Ω = 1,F0 = 2.00, ω = 0, ϵ = 0.01 (5.1)

ζ(s) = 0.25, σ(s) = 0, γ = 0.50, µ(s) = −10 + 20 s, (5.2) so that the individual mass resonators are uniformly distributed over the interval

[0.90, 1.10]. Finally the discrete population is uniformly distributed so that

i − 1 1 ∑N s = , ρ(s) = δ(s − s ). (5.3) i N − 1 N i i=1

68 5.1 Monostable-Unique Solutions

As seen in the Chapter 4, for certain values of the coupling parameters (α, β) and nonlinearity, the global coupling leads to an effective forcing G⋆ on each resonator that is identical across the population. For G⋆ sufficiently small the resulting resonator response is single-valued leading to a unique equilibrium distribution.

At coupling values of (α, β) = (−1.0, −2.0), only an unique solution is obtained by solving for A(si) ,B(si) in Eq. (3.22), amplitude response is shown in

Figure 5.1(a). Here it can be seen that all resonators have an unique solution, with the global self-consistency constants identified as (V,W ) = (0.235, −0.1813) which satisfy Eq. (3.24), and shown in Figure 5.1(b). The response in Figure 5.1(a) shows a distribution in which resonator i = 7 on branch X(s) > 0 and resonators succeeding index are on X(s) > 0 and resonators with preceding index numbers are on X(s) < 0, hence it is denoted as E7 (this notation was discussed in Chapter 3).

69 8 1

6 0

4 W R(s) -1 2

0 -2 0.9 0.95 1 1.05 1.1 -1 0 1 2 m(s) V

Figure 5.1: a) Equilibrium distribution showing states of resonators(R(s) vs m(s)) b)

Global self consistency constants (V,W ) = (0.235 , −0.1813), at the coupling values of (α , β) = (−1.0 , −2.0), only an unique solution state exists, bc bc , E7 indicates the state of individual resonators in the equilibrium distribution.

70 5.2 Multiple Equilibrium Distributions

As the coupling parameters (α, β) vary the effective forcing G⋆ increases, leading to multi-valued resonator curves. The bi-stability exhibited by individual resonators can lead to additional equilibrium populations, so that for a given set of parameter values multiple equilibrium distributions may exist.

The equilibrium states of resonators are dependent on dissipative and reactive coupling, it was introduced in the Chapter 3 different states can be observed by varying them. In this section a detailed study of various possible states and their stability behavior is discussed in detail. Initially we start at a constant dissipative coupling of α = −1.0 and vary reactive coupling β to find out the possible distributions, explain the bifurcations that occur. Then we discuss this phenomenon at different α eventually.

5.2.1 Bifurcations

The effect of coupling parameters on various distributions, needs to be discussed, especially leading to multiple equilibrium distributions by varying the couple parameters. In an uncoupled system (or in case of a Forced Duffing resonator), the two saddle-node bifurcations that occur are at upper and lower inflection points of the continuity response curve. Here in the case of a large array of resonators the associated bifurcations are not directly described by the bifurcations seen in the underlying resonator curve, but are related.

71 8 3.5

4 3 R(s) R(s)

2 2.5 0 0.9 0.95 1 1.05 1.1 1.03 1.032 1.034 1.036 1.038 m(s) m(s)

Figure 5.2: a) Equilibrium distributions after bifurcation, referred as Type-I saddle- node bifurcation points for coupling parameters (α , β) = (−0.3 , 1.2) b) Enlarged

bc bc E U bc bc E U image showing resonator s7 on same branches 7 (unstable), 7 .

The bifurcation that occur near upper inflection point would be called as

Type-I bifurcation, this occurs when a resonator is near to upper critical point lead-

E U E ing to two distributions of same type. These distributions can transit to i and i+1 distribution when the coupling parameter are further changed from its value at bifurcation. Figure 5.2 shows two distributions that resulted from this bifurcation due to resonator s7 nearer to upper inflection point and two distributions that resulted

E U after bifurcation are 7 .

Type-II bifurcation occurs at the lower inflection point when the two distributions coalesce or emanate after the resonator reaches near to lower inflection point, at the point of bifurcation two distributions of same type arise. These distributions can

E E U transit to i and i distributions when the coupling parameter are further changed from its value at bifurcation. Here in Figure 5.3 two distributions shown here are

72 8 3.5

6 3

4 2.5 R(s) R(s) 2 2 1.5 0 0.9 0.95 1 1.05 1.1 1.072 1.076 1.08 m(s) m(s)

Figure 5.3: a) Equilibrium distributions after Type II bifurcation for coupling pa-

bc bc E bc bc E U rameters (α , β) = (0.1 , 5.3) b) Enlarged image of resonator, 9, 9 .

E E U 9 and 9 , when chosen coupling parameters are slightly away from its bifurcation value, this bifurcation occurs when resonator s9 is near to the lower inflection point.

5.2.2 Fixed α = −1.00

For fixed α = −1.00 varying the dissipative coupling parameter β leads to resonator curves that exhibit bi-stablity for individual resonators. As a result isolated branches of equilibrium arise from saddle-node bifurcations (Type I) associated with individual resonators, and both the existence and stability of these solutions are described in terms of the resonator curves and self-consistency variables as β varies. For α =

−1.00, the additional equilibrium populations that are involved in the bifurcations are primarily differentiated by the response of resonator s7.

In the previous section only equilibrium state at (α , β) = (−1.0 , −2.0) is shown, keeping α unaltered when the reactive coupling is changed to (α , β) =

73 8 1

6 0

4 W R(s) -1 2

0 -2 0.9 0.95 1 1.05 1.1 -1 0 1 2 m(s) V

Figure 5.4: a) Equilibrium distributions showing states of resonators(R(s) vs m(s))

bc bc E bc bc E bc bc E U 7, 8 and 7 , b) Global self consistency (V,W ), at the coupling values of (α , β) = (−1.0 , 2.0).

(−1.0 , 2.0), it can be observed from the Figure 5.4(a) that two additional new equilibrium states emerge. The Eq. (3.22) and Eq. (3.24) satisfy for two new sets of global self-consistency (V,W ) variables as seen in Figure 5.4(b). The emergence of these solutions can be attributed to, the resonator i = 7 exhibiting bi-stability as seen in the

Figure 5.4, two additional distributions can be seen. The distribution indicated by

bc bc has resonator i = 8 on branch X(s) > 0 and resonators succeeding index are on

X(s) > 0 and resonators with preceding index numbers are on X(s) < 0 are named as E8. The distribution shown using bc bc has resonator i = 7 on unstable branch of X(s) > 0 and resonators succeeding index are on X(s) > 0 and resonators with

E U preceding index numbers are on X(s) < 0 are named as 7 .

The stability of an equilibrium state can be obtained by solving for the discrete spectrum, λ in Eq. (3.32), number of eigenvalues for the given system of res-

74 onators (N = 10) is twenty and its sign of real part for each eigenvalue decides the stability. Equilibrium states E7, E8 have the eigenvalues which have real parts less

E U than zero hence they are stable as shown in Figures 5.5a, 5.5c, 7 has an eigenvalue greater than zero, hence it is unstable as shown in Figures 5.5b.

Global self-consistency variables (V,W ) as a function of β are shown in Fig- ure 5.6, these influence the behavior of these distributions, which in turn aremainly

′ driven by behavior of resonator s7 (here). The P and P indicate point of saddle- node bifurcations at Type-I. The detailed reasoning for such bifurcations of the new branches has been explained in rest of this section.

′ The equilibrium distributions at these points P and P are shown here, these are when β = −0.06 and β = 3.07 respectively. Near the point P two equilibrium states are shown in Figure 5.7 this is the point of Type-I bifurcation, here two equi-

E U librium states of type 7 exist, it can be seen from Figure 5.7b that in these type the resonator s7 is on the unstable branch. In Figure 5.8 equilibrium distributions near

′ the point P are shown where the two equilibrium states are of E8, the enlarged picture in Figure 5.7b shows the resonator s7 in both these distributions is on X(s) < 0 branch, and the bifurcation occurs when the resonators this resonator in these two different distributions coalesce.

The response of resonator s7 in different distributions can be shown in Fig- ure 5.9(a), in case of distribution E7, the response increases gradually with increase in β, the point P which as mentioned earlier is a point of bifurcation of Type-I,

E U where two distributions of 7 emanate, increasing β further shifts the resonator s7

75 10 10

5 5 ) ) 0 0 Im( Im( -5 -5

-10 -10 -1 0 1 -1 0 1 Re( ) Re( )

bc bc E bc bc EU (a) Discrete Spectrum, 7 (b) Discrete Spectrum, 7

5 ) 0 Im( -5

-10 -1 0 1 Re( )

− E E U Figure 5.5: Spectrum plot Re(λ) vs Im(λ), (α , β) = ( 1.0 , 2.0) a) 7, b) 7 and c)

E E E E U 8 , both 7 and 8 have eigenvalues with Re(λ) < 0; 7 has eigenvalue Re(λ) > 0

76 0.5 0

-0.2 P V

0 W -0.4 P P' -0.6 P' -0.5 -0.8 -2 0 2 4 6 -20 2 4 6

(a) V vs β (b) W vs β

-0.5 W P P' -1 0.5 6 4 0 2 0 -0.5 -2 V

Figure 5.6: Self-Consistency variables (V,W ) as a function of reactive coupling (β)

− E E E E U at α = 1.0, 7, 8, 8(unstable) , 7 , Figure (a) and (b) shows two

′ different orthogonal planes of Figure (c), points P and P indicate Type-I bifurcation

77 8 3 6

R(s) R(s) 2.95

2 2.9 0 0.9 0.95 1 1.05 1.1 1.033 1.0335 1.034 m(s) m(s)

(a) R(s) vs m(s) (b) Enlarged image of (a)

bc bc E U Figure 5.7: Two similar distributions of 7 near the point P (α , β) =

(−1.0 , 0.06), enlarged picture here shows resonator 7 in two distributions, indicating

Type-I bifurcation.

78 8 4.5 6 4 4

R(s) R(s) 3.5 2 3

0 0.9 0.95 1 1.05 1.1 1.03 1.04 1.05 1.06 1.07 m(s) m(s)

(a) R(s) vs m(s) (b) enlarged image of (a)

′ Figure 5.8: Two similar distributions of bc bc E8 near the point P (α , β) =

(−1.0 , 3.06), enlarged image here shows resonator 7 in two distributions, indicating Type-I bifurcation.

79 4 3.4

3 P' 3.2 ) ) 7 7 2 R(s R(s Q 3 ' Q 1 P

0 2.8 -2 0 2 4 6 -1 0 1 2 3 4

(a) R(s7) (b) enlarged image of (a)

E E U Figure 5.9: Response of resonator s7 (R(s7) vs β) for the distributions 7, 7 ,

′ ′ E8, E8(unstable), Q and Q indicate the branch change points, P and P indicate the points of Type-I bifurcation.

E U E on to the branch X(s) < 0 transforming one equilibrium distribution to 7 to 8, this branch transform point is indicated by the point Q. Any further increase in β leads to increase in response for this distribution, while the response of resonator s7

E U in other distribution 7 continuously decreases until the resonator s7 transforms to

′ E8(unstable) point where β is at Q , then the newly transformed distribution continuously increases and coalesces at upper-saddle node.

80 The relationship between self-consistency variables and global coupling force

G∗ on β are discussed here, in multiple distributions each individual resonator can have X(s) > 0 to either unstable part of X(s) > 0 branch or to a X(s) < 0 branch.

The amplitude R(s) and X(s) of each isolated resonators due to increase in β are also shown here. The coupling parameters (α, β) and the global self consistency constants (V,W ) are related to the response of system is from Eq. (5.4) (as obtained in Chapter 3)

G2 R2(s) = ⋆ , (5.4) X2(s) + (2ζ(s)Ω2)2 where

G1 ≡ α Ω V + β W, G2 ≡ −F0 + α Ω W − β V, (5.5) 3γ X(s) ≡ Ω [(µ(s) − σ(s)) Ω − 2ω] − R2(s). (5.6) 4 and

( )( ) 2 ≡ 2 2 ≡ 2 − − 2 2 2 2 G⋆ G1 + G2 F0 2 (αΩ W β V ) F0 + (αΩ) + β V + W . (5.7)

Rewriting the Eq. (5.4) in terms of X(s), we obtain

( ) 3 γ X2(s) + (2 ζ(s)Ω2)2 (X(s) − D(s)) + G2 = 0, (5.8) 4 ⋆

where D(s) = (µ(s) − σ(s)) Ω − 2ω. X(si) can have positive or negative values and these combination leads to different distributions, but the magnitude and sign of

X(si) is dependent on D(si) and G∗. Here detuning factor D(si) varies for the each resonator in the population, whereas G∗ is same for whole population of resonators, 81 4 1

0.5 2 ) ) P 7 7 Q' 0 Q X(s X(s 0 -0.5 P'

-2 -1 -2 0 2 4 6 -1 0 1 2 3 4

(a) X(s7) (b) enlarged image of (a)

E E U E E Figure 5.10: X(s7) vs β for equilibrium distributions 7, 7 , 8, 8

′ (unstable), points P and P indicate the bifurcation of Type I, Q is the point where the

E U E ′ distribution 7 transforms to 8(stable), while Q is the point where the distribution

E U E 7 transforms to 8 (unstable).

but G∗ is dependent on (V , W , α , β), assuming F0 doesn’t change. X(s7) as a function of β for various equilibrium distribution can be shown in Figure 5.10, it

E U E is not influenced much in case of distribution X7 but 7 and 8 are considerably

′ effected by change in β. As discussed previously, points P and P indicate Type I bifurcation, in that coupling range and Q is the point where the resonator s7 in one of

E U the two distribution 7 changes its branch from X(s) > 0 to X(s) < 0 thus making

′ a distribution E7(Stable), while Q is such transformation which does the same but

E8 (unstable) in this case.

E U E The distributions of 7 and 8 are shown at different values of β in the

82 E U Figure 5.11. As explained previously, two distributions of 7 type appear at point

P when β = −0.06, then with slight increase in β, resonator s7 in one of these distributions shifts from X(s) > 0 to X(s) < 0, this gives birth to E8 distribution.

At the value of β = 0.5, the two distributions that exist is shown in Figure 5.11a, by increasing the β, amplitude of resonator in s7 distribution increases in distribution E8

E U and decreases in 7 , this is evident from Figure 5.9, the same can also be seen when

E U β = 1.5 from the Figure 5.11c. For β > 2.2 even in the distribution 7 the response of resonator s7 increases and with the further increase in β the resonator s7 in the

E U 7 distribution, changes its branch from from X(s) > 0 to X(s) < 0.

E U E This results in transforming distribution 7 to 8 as can be seen in Fig- ure 5.11c at β = 2.5, but this distribution is not stable, the stability states for these coupling values can be seen in Figure 5.12. With further increase in β the amplitude of resonator s7 in the newly transformed distribution increases at a faster rate than the same resonator s7 in the previously existing distribution, but the amplitude response of existing distribution is greater than the newly transformed distribution. Finally, these two distributions coalesce and disappear at β > 3.06 indicating bifurcation of

Type-I.

5.2.3 Variations in α

E U E As α varies, the isolated branch that gives rise to 7 and 8, observed previously, deforms. Additional branches appear through the same mechanism, but with different resonators involved in the bifurcation. These new branches are again described in

83 8 8

6 6

4 4 R(s) R(s) 2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

(a) β = 0.5 (b) β = 1.5

8 8

6 6

4 4 R(s) R(s) 2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

E U E Figure 5.11: Response of equilibrium distributions, 7 , 8, they transform with variation in β at α = −1.0.

84 10 10

5 5 ) ) 0 0 Im( Im( -5 -5

-10 -10 -1 0 1 -1 0 1 Re( ) Re( )

(a) Stable Distribution (b) Unstable Distribution

Figure 5.12: Spectrum plot for (α , β) = (−1.0 , 2.5) a) Equilibrium state E8 is stable, b) Newly transformed equilibrium state E8 is unstable.

terms of the self-consistency variables (V,W ), as well as the global coupling force G⋆ and are identified by the effective detuning X(si).

Type-I Bifurcation on resonator s7

As seen in the previous subsection, at α = −1.0 and varying β resulted in the bifurcation of Self-Consistency variables (V,W ). This clearly indicates that the effect of β on a distributions is not only to alter its amplitude response through G∗, but also initiate the bifurcations. Similarly affect of α on these distributions is shown here. Figure 5.13 shows the effect of α on self-Consistency variables (V,W ), while its relation on G∗ is shown in Figure 5.14. At α = −1.5 there is only one equilibrium state E7 as β is varied while the sign of V changes from positive to negative as β is

85 increased, but W doesn’t change its sign as seen in Figure 5.13a. G∗ at α = −1.5 in Figure 5.14a continuously increases till V changes it sign from positive to nega-

− E E U tive. At α = 1.2 in Figure 5.13b, two equilibrium states transit to 8 and 7 after

β crosses a critical value. As explained in the previous subsection, the equilibrium

E U E state 7 transforms to 8(unstable equilibrium state) and finally merge forming a close contour. While the V changes its sign for both the distributions from positive to negative, sign of W remains negative throughout for both.

This is the reason for behavior of G∗ seen in Figure 5.14b, it increases with increase in β as long as V has a positive sign, while it decreases when V has a negative sign. At α = −0.6( Figures 5.13c, 5.14c) and α = −0.3 (Figures 5.13c, 5.14d) the branches emerge at a lower values of β and end at higher values than previously seen.

Thus the effect of α is to scale up or inflate the size of closed contours and influence the bifurcation points.

Type-I bifurcation on resonator s8

Similar to resonator s7, even the resonator s8 exhibits bi-stability as seen in Fig-

E U E ure 5.15. The two distributions transit to 8 and 9 after Type-I bifurcation on s8 .

As β is further increased, the amplitude of resonator s8 in E9 increases while the am-

E U plitude of resonator s8 in 8 decreases. This trend continues until β = 1.124 and then

E U the amplitude of s8 in unstable distribution 8 also increases. At value of β = 1.737 the resonator s8 changes its branch from X(s) > 0 to X(s) < 0, now we have two distributions indicated by E9, which have different amplitudes for resonators in their

86 (a) α = −1.5 (b) α = −1.2

Figure 5.13: Self-Consistency variables (V,W ) as a function of reactive coupling (β)

E E E E U at different α, 7, 8, 8(unstable) , 7 , all the bifurcations that are shown here are of Type-I

87 (a) α = −1.5 (b) α = −1.2

Figure 5.14: Global coupling force G∗ as a function of reactive coupling (β) at different

E E E E U α, 7, 8, 8(unstable) , 7 , all bifurcations are Type I.

88 8

4 R(s)

0 0.9 0.95 1 1.05 1.1 m(s)

E E U − Figure 5.15: Response for 9 , 8 at (α , β) = ( 0.3 , 1.0)

population. The rate of increase in amplitude for resonator s8 in newly transformed distribution of E9 is slightly greater than the previous distribution of E9 and both of these distributions coalesce at β = 3.389 and disappear for further increase in β saddle node bifurcation of Type-I. Amplitude R(s8) and X(s8) as a function of β in various distributions are shown in the Figure 5.16.

Type-II Bifurcation on resonator s9

As discussed in the section of bifurcations, the Type-II bifurcation occurs at Lower- saddle node, it can be demonstrated by varying α. The amplitude of resonator s8 in

E E U distributions at varying α is shown in Figure 5.17, at α = 0 distributions 9, 8 , can be seen in Figure 5.17a, but with increase of α to 0.15 an additional distribution that can be seen in Figure 5.17b, this occurs when resonator s9 crosses to the unstable

E U branch of continuous curve. The region of 9 further increases with increase in α as seen in Figure 5.17c. This type of bifurcation can be seen in Figure 5.17b where the

89 4.4 0.5

4.2 0 ) ) 8 8 4 -0.5 X(s R(s 3.8 -1

3.6 -1.5 0 1 2 3 4 0 1 2 3 4

(a) R(s8) vs β (b) X(s8) vs β

E U E Figure 5.16: Equilibrium distributions represented by, 8 , 9(unstable),

E9, α = −0.3 a) Amplitude of resonator R(s8) vs β, b) X(s8) vs β

resonator s9 is present on both stable-unstable branches.

The Global Self-Consistency variables (V,W ) and G∗ as a function of β at different α are shown in Figure 5.18 and Figure 5.19 respectively. At α = −0.45 the equilibrium states that occur are both E9 (but one stable and unstable states), the closed contour of V,W is very small as seen in Figure 5.18a, just like seen in previous case of resonator s7 exhibiting bi-stability, even here the two different equilibrium states with increase in β, V changes from positive to negative and where as W remains negative all through. For the scale seen in the Figure 5.19a, G∗ appears to be a small point, it follows the same path as seen previously. At α = −0.2 in

E U Figure 5.18b the branch of 8 can be seen, this was expected at higher α as explained previously and force G∗ seen in Figure 5.19b increases until V changes sign and then

90 5 5

4.5 4.5 ) ) 8 8

R(s 4 R(s 4

3.5 3.5 0 2 4 6 8 0 2 4 6 8

(a) α = 0 (b) α = 0.15

4.5 ) 8

R(s 4

3.5 0 2 4 6 8

Figure 5.17: Amplitude of resonator s8 R(s8) vs β for equilibrium distributions

E U E E E U 8 , 9, 9(unstable), 9 , at different α

91 decreases. With increase to α = 0 the contours previously seen are scaled up as seen in Figure 5.18c, so does the G∗ that is seen in Figure 5.19c. Further increase to α = 0.2, as resonator s9 changes from stable part of X(s) > 0 to unstable part

E E U of continuous Duffing curve in 9 distribution transforms to 9 , this can be seen in

Figure 5.20, the newly transformed distribution is unstable. This transformation is caused as G∗ influences the resonator s9 to shift the branch. Thus Self-Consistency variables (V,W ) as a function of β as seen in Figure 5.18d and Figure 5.19 has the

E U E E U branch showing equilibrium distribution 9 and finally two branches of 9 and 9 coalesce. Thus α not only scales up the branches of equilibrium distributions, but would further introduce new branches in the closed contours.

Type I Bifurcation on resonator s10

Even the resonator s10 exhibits bi-stability as seen in Figure 5.21, here two equilibrium states of E∞ can be seen where one is stable and the other being unstable, the other

E U E equilibrium state observed here are 10. The equilibrium distributions transit to ∞

E U and 10 after β > 0.4455, due to Type-I bifurcation on resonator s10. Amplitude

E U response of s10 in the distribution 10 decreases with increase in β until it reaches

β > 0.742, after that the response amplitude increase with β, where as amplitude response of s10 in the distribution E∞ continuously increases, as seen in Figure. 5.22.

E U The resonator s10 in the distribution 10 changes its branch from X(s) > 0 and

X(s) < 0 at β = 1.101, thereby giving rise to newly transformed distribution of E∞ type. Now in both of these distributions with increase in β the response of resonator

92 (a) α = −0.45 (b) α = −0.2

Figure 5.18: Self-Consistency variables (V,W ) as a function of reactive coupling (β)

E U E E E U at different α, 8 , 9, 9 (unstable), 9 , both Type-I and Type-II can be seen here.

93 (a) α = −0.45 (b) α = −0.2

Figure 5.19: Global coupling force G∗ as a function of reactive coupling (β) at different

E U E E E U α, 8 , 9, 9 (unstable) 9 , both Type-I and Type-II can be seen here

94 8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

(a) β = 6.8 (b) β = 7.0

Figure 5.20: At α = 0.2, the equilibrium distributions influenced by resonator s8 are

bc bc E bc bc E U E 9, 9 , a) at β = 6.8 two of the distributions are of 9 type, b) at higher β resonator s9(from stable distribution of E9) changes from stable part of X(s) > 0 to

E E U unstable part of continuous Duffing curve in 9 distribution transforms to 9 .

95 8

4 R(s) 2

0 0.9 0.95 1 1.05 1.1 m(s)

E E U Figure 5.21: Response for ∞ , 10 at (α , β) = (0.08 , 2.0)

s10 would go up and they finally coalesce at β = 6.971 and disappear for any larger

β.

Similar to resonator s8, even the resonator s10 at α = −0.05, has two equilibrium states of type E∞(stable and unstable states), Self-Consistency variables V,W as a function of β and G∗, shown in Figures 5.23a and 5.24a also show the same.

Increase in α = 0 to a higher value would not only scale up the closed contour of

V,W as shown in Figure 5.23b, but also increase the coupling force G∗ as seen in

E U Figure 5.23d. By changing α = 0.1 would introduce the distribution 10 in the contour as seen in Figure 5.23c and coupling force increases with change in β until V changes sign from positive to negative, this can be seen in Figure 5.24c. The change in the coupling force with change in α is much greater when resonator s10 exhibits bi-stability, this can be seen in as Figure 5.24 curves of G∗ as a function of β become steeper.

96 6.5 2

0 ) 6 ) 10 10 -2 X(s R(s 5.5 -4

5 -6 0 2 4 6 8 0 2 4 6 8

(a) R(s10) vs β (b) X(s10) vs β

E U E E Figure 5.22: Equilibrium distributions represented by, 10, ∞, ∞ (unstable) a) Amplitude of resonator R(s10) vs β, b) X(s10) vs β at α = 0.08, Type I bifurcation

97 (a) α = −0.05 (b) α = 0

Figure 5.23: Self-Consistency variables (V,W ) as a function of reactive coupling (β)

E U E E at different α, 10, ∞, ∞ (unstable), Type I bifurcation.

98 (a) α = −0.05 (b) α = 0

Figure 5.24: Global coupling force G∗ as a function of reactive coupling (β) at different

E U E E α, 10, ∞, ∞ (unstable), Type-I bifurcation can be seen here.

99 (a) (V,W ) vs β (b) (V,W )

Figure 5.25: Self-Consistency variables (V,W ) as a function of reactive coupling (β) at α = 0.08, a) 3-D plot of (V,W ) vs β b) 2-D plot of (V,W ), Open Contour ( E7),

E U E E E U Closed Contour-1 ( 7 , 8, 8 (unstable)), Closed Contour-2 ( 8 ,

E E E U E E 9, 9 (unstable)), Closed Contour-3 ( 9 , 10, 10 (unstable)) ,

E U E E Closed Contour-4 ( 10, ∞, ∞ (unstable)).

Coexisting Isolated Equilibrium Branches

As explained above the effect of α scales up the closed contours and β introduces the bifurcation in Figure 5.25 at α = .08 some contours are named as shown, where open contour corresponds to an isolated equilibrium state E7). The other contours

E E U are closed ones, where each of them have equilibrium states i , i−1(i = 8 ... 10).

As explained previously at lower values of α there were only two equilibrium states that exist in each contour, but higher the value of α leads to one more equilibrium state in the closed contour.

100 5.3 Non-isolated Equilibrium states

For the previously considered values of α, the isolated branches of equilibrium arose with resonator i − 1 near the upper saddle-node bifurcation present in the resonator

E E U response, so that the branch was to equilibrium states ( i , i−1). However, as α is further increased resonator i can move onto the unstable branch of the resonator

E U response, so that the equilibrium branch can expand to include distributions i , as seen in the Figure 5.20, which indicates Type II bifurcation.

Global-self consistency variables (V,W ) for different equilibrium states at various α are discussed in the previous section. Each of those closed curves(called

E E U contours) indicating by i , i−1(i = 8 ... 10). But each of these contours are no more just a collection of two states but there will be a third equilibrium state gets introduced with further increase in α. When the coupling value is relatively low,

E E U the global self-consistency constant α = 0.08 are isolated branches of i , i−1(i =

8 ... 10), the set of such branches are shown in Figure 5.25. As α is further increased the new patches emerge in these isolated branches, following example shows how they emerge, on a Closed Contour 3, Figure 5.26 shows global coupling force as a function

1 5 of β. C3 to C3 indicate discrete points on the contour, it is used to indicate various equilibrium states as β is varied.

E 1 The equilibrium states 10 at C3 occurs at β = 2.0 as shown in Figure 5.27a,

2 increase in β leads to increases G∗ and at the point C3 where β = 5.5 the amplitude response of equilibrium state E10 scales up as seen in the Figure 5.27b. When equi-

101 E U E E Figure 5.26: G∗ as a function of β in Closed Contour-3 ( 9 , 10, 10

(unstable)) at α = 0.08

3 librium distribution at a point C3 on the unstable part of closed contour that is at a value of β = 6.0, G∗ is less and the equilibrium state is unstable Figure 5.27c. As

4 equilibrium distribution at point C3 is taken when β = 3.5, there is further decrease in G∗ and overall amplitude response as seen in Figure 5.27d. As the equilibrium

5 point C3 is considered, it can be observed that the resonator s9 changes the branch

E U X(s) < 0 to unstable part of X(s) > 0 resulting in the equilibrium state 9 as

1 shown in Figure 5.27e. The global coupling force in case of C3 is shown in enlarged

Figure 5.27f.

E U When α = 0.2 the Closed Contour-3 has a patch of equilibrium 10 as seen in Figure 5.28, with further increase in α this patch gets increased. The equilibrium states that occur for various points along the closed contour are shown in Figures 5.29

1 and Figure 5.30. At the point C3 for β = 2.0 the only equilibrium state that exists are

E 2 as shown in Figure 5.29a where 10, further increase in β = 5.5 at the point C3 leads

102 8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

1 2 (a) β = 2.0 at point C3 (b) β = 5.5 at point C3

8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

3 4 (c) β = 6.0 at point C3 (d) β = 3.5 at point C3

103 8

4 R(s)

0 0.9 0.95 1 1.05 1.1 m(s)

5 5 (e) β = 0.5 at point C3 (f) G∗ near the region C3 .

Figure 5.27: Different equilibrium distributions that occur for discrete points from

1 5 bc bc E U bc bc E C3 to C3 and on the curve shown in Figure 5.26 at α = 0.08, 9 , 10.

to increase in G∗ also leading to increase in overall amplitude as seen in Figure 5.29b.

With increase in β further at this point would lead to introduction of new patch as being discussed here, in the closed contour, at this α, the resonator s10 in the existing equilibrium state would change from X(s) > 0 stable to unstable part, decrease in

β leads to decrease in G∗ as this transformation occurs for β = 6.3 the amplitude response can be shown in Figure 5.29c. Further increase in β would lead to increase in G∗ leading to the equilibrium state Figure 5.29d. The global coupling force in case

5 of Contour 3 is shown in Figure 5.29e. As the point C3 is considered the resonator s10 is on the stable part of X(s) > 0 as shown in Figure 5.30a, further decrease in β leads to decrease in G∗ thus decreasing in the overall response as shown in Figure 5.30b.

Decreasing β further leads to decrease in overall response and resonator s9 shifts

104 E U E E Figure 5.28: G∗ as a function of β in Closed Contour-3 ( 9 , 10, 10

E U (unstable), 10) at α = 0.2

7 branch X(s) < 0 to X(s) > 0 and the response at C3 for β = 1.0 as shown in

Figure 5.30c.

Global self-consistency variables at α = 0.2 as function of β are shown in

Figure 5.31, where the contours have additional patches as explained previously are as shown in Figure 5.31, G∗ as a function of β are shown in Figure 5.32.

105 8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

1 2 (a) β = 2.0 at point C3 (b) β = 5.5 at point C3

4 R(s)

0 0.9 0.95 1 1.05 1.1 m(s)

3 (c) β = 6.3 at point C3

Figure 5.29: Different equilibrium distributions that occur for discrete points from

1 4 bc bc E U bc bc E C3 to C3 and on the curve shown in Figure 5.28 at α = 0.2, 10, 10

106 8

4 R(s)

0 0.9 0.95 1 1.05 1.1 m(s)

4 (d) β = 6.5 at point C3 (e) Enlarged G∗ in Contour 3

Figure 5.29: Different equilibrium distributions that occur for discrete points from

1 4 bc bc E U bc bc E C3 to C3 and on the curve shown in Figure 5.28 at α = 0.2, 10, 10(cont.)

107 8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

5 6 (a) β = 6.0 at point C3 (b) β = 3.5 at point C3

4 R(s)

0 0.9 0.95 1 1.05 1.1 m(s)

7 (c) β = 1.0 at point C3

Figure 5.30: Different equilibrium distributions that occur for discrete points from

5 7 bc bc E U bc bc E C3 to C3 and on the curve shown in Figure 5.28 at α = 0.2, 9 , 10

108 (a) ((V,W ) vs β (b) (V,W )

Figure 5.31: Self-Consistency variables (V,W ) as a function of reactive coupling

(β) at α = 0.2, (a) 3-D plot of (V,W ) vs β b) 2-D plot of (V,W ), Open Con-

E E E U E U E E tour ( 7, 6, 7 ), Closed Contour-1 ( 7 , 8, 8 (unsta-

E U E U E E E U ble), 8 ), Closed Contour-2 ( 8 , 9, 9(unstable), 9 ), Closed

E U E E E U E U Contour-3 ( 9 , 10, 10 (unstable), 10), Closed Contour-4 ( 10,

E∞, E∞ (unstable)).

109 (a) G∗ vs β (b) Enlarged

Figure 5.32: Global coupling force G∗ as a function of reactive coupling (β) at α = 0.2

E E E U E U E E Open Contour ( 7, 6, 7 ), Closed Contour-1 ( 7 , 8, 8

E U E U E E E U (unstable), 8 ), Closed Contour-2 ( 8 , 9, 9(unstable), 9 ),

E U E E E U Closed Contour-3 ( 9 , 10, 10(unstable), 10) , Closed Contour-4

E U E E ( 10, ∞, ∞(unstable)).

110 E U E E Figure 5.33: G∗ as a function of β in Closed Contour-3 ( 9 , 10, 10

E U (unstable), 10) at α = 0.3

5.3.1 Merging of Isolated Branches

As described in the previous section, for sufficiently large α the isolated equilibrium

E E U E U branches that arise localized to distributions i and i−1 can expand to include i .

E U E U These then coexist with additional branches that arise from i and i . For further

E U increases in α the two equibrium branches that contain i can merge in to a single entity.

As seen previously, α not only scales up the closed contours of global self- consistency constants V,W but also introduces the new patches in the closed contours, further increase in α merges these isolated branches and it will be discussed here. Figure 5.33 shows G∗ as a function of β at α = 0.3, where different discrete

1 points in the closed contour, for a point C3 where β = 2.5 the equilibrium distribution

E10 can be seen in Figure 5.34a.

2 Increase in β leads to increase in G∗ and at the point C3 the amplitude

111 response at β = 5.0 is shown in Figure 5.34b. Initially increasing β would lead to increasing G∗ until a point where the resonator s10 changes the branch from unstable part of X(s) < 0 to X(s) > 0, this results in new equilibrium state as explained in

3 the previous subsection, the equilibrium state at point C3 for β = 6.0 is shown in

4 Figure 5.34c. On these contour as point C3 is taken for the consideration response of the distribution where β = 4.5 and the amplitude response seen in Figure 5.34d has reduced to due to decrease in G∗ compared to earlier contour point. As the point

5 C3 is taken where the amplitude response as shown in Figure 5.35a at β = 7.0, due

6 7 to increase in G∗ the amplitude response has increased. As the points C3 , C3 are considered where β = 5.75 and β = 3.5, these distributions are unstable. The points

8 9 C3 , C3 are shown where the resonator s9 has shifted from X(s) < 0 to unstable part

E U of X(s) > 0, leading to equilibrium distribution 9 as shown in Figure 5.35d and

5.35e.

Similar exercise of plotting the distributions along the Closed Contour-4 is shown in Figure 5.36, the enlarged picture with discrete point on this contour are

1 2 shown in Figure 5.36b. The points C4 and C4 represent the equilibrium state, where due to increase in G∗ with increase in β leads to increase in overall amplitude response

3 4 as shown in Figures 5.37a and 5.37b. Similarly for the points C4 and C4 on Closed

Contour-4, representing the unstable E∞, it is obvious that with decrease in β leads to decrease in G∗ leading to scaling down of amplitude, as seen in Figure 5.37e and

5 5.37e at β = 6.0 and β = 2.5 respectively. A point C4 represents the equilibrium distribution where resonator s10 shifts to unstable part of X(s) > 0 from X(s) < 0

112 8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

1 2 (a) β = 2.5 at point C3 (b) β = 5.0 at point C3

8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

3 4 (c) β = 6.0 at point C3 (d) β = 4.5 at point C3

Figure 5.34: Different equilibrium distributions that occur for discrete points from

1 4 bc bc E bc bc E U C3 to C3 and on the curve shown in Figure 5.33 at α = 0.2, 10, 10.

113 8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

5 6 (a) β = 7.0 at point C3 (b) β = 5.75 at point C3

4 R(s)

0 0.9 0.95 1 1.05 1.1 m(s)

7 (c) β = 3.5 at point C3

5 Figure 5.35: Different equilibrium distributions that occur for discrete points from C3

9 bc bc E U bc bc E U bc bc E to C3 and on the curve shown in Figure 5.33, at α = 0.2 10, 9 , 10.

114 8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

8 9 (d) β = 2.0 at point C3 (e) β = 0.5 at point C3

5 Figure 5.35: Different equilibrium distributions that occur for discrete points from C3

9 bc bc E U bc bc E U bc bc E to C3 and on the curve shown in Figure 5.33, at α = 0.2 10, 9 , 10

(cont.).

E U leading to an equilibrium distribution 10 as shown in Figure 5.37e at β = 0.5.

With increase in α, the Closed Contour-3 and Closed Contour-4 coalesce, this can be evidenced from Figure 5.38 and 5.39. The patch of E10 increases with increase in α as the resonator s10 from X(s) > 0 to unstable part of X(s) > 0. Similarly the

Closed Contours-4 has increase in the patch of E10 with increase in α. Increase α would further increases the patch of E10 an both the contours and finally results in coalescing the contours and at α = 0.38, there is only a single contour.

Increasing α further leads to merging of contour 2 with 3 as can be seen

E U in Figure 5.40, as explained above even here the branch representing 9 on two of these contours merge resulting in a single contour as can be seen in Figure 5.40b.

115 (a) Closed Contour-4 (b) enlarged image of (a)

E U E Figure 5.36: G∗ as a function of β in Closed Contour-4 ( 10, ∞,

E∞(unstable)) at α = 0.3

Similarly at higher α merging of Closed Contour-1 with Closed Contour- 2 can be

E U seen at α = 0.62, where 8 on each of these branches merge forming a single contour as seen in Figure 5.41b.

At higher α merging of Open Contour with Closed Contour-1 can be seen at

E U α = 0.86, where 7 on each of these branches merge forming a single contour as seen in Figure 5.42b. Figure 5.43a shows the global coupling force (G∗) and global self- consistency variables (V,W ) are shown in Figure 5.43b as β is varied at α = 0.86, here there are no isolated contours as seen at lower values of α, a continuous curve of both coupling force and consistency variables clearly indicating all the coupled equilibrium states are dependent on each other.

116 8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

1 2 (a) β = 1.5 at point C4 (b) β = 7.0 at point C4

4 R(s)

0 0.9 0.95 1 1.05 1.1 m(s)

3 (c) β = 6.0 at point C4

Figure 5.37: Different equilibrium distributions that occur for discrete points from

1 5 bc bc E U bc bc E C4 to C4 and on the curve shown in Figure 5.36 at α = 0.3, 10, ∞.

117 8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

4 5 (d) β = 2.5 at point C4 (e) β = 1.0 at point C4

Figure 5.37: Different equilibrium distributions that occur for discrete points from

1 5 bc bc E U bc bc E C4 to C4 and on the curve shown in Figure 5.36 at α = 0.3, 10, ∞ (cont.).

118 (a) α = 0.30 (b) α = 0.32

E U E E Figure 5.38: Enlarged pictures of Closed Contour-3 ( 9 , 10, , 10 (unsta-

E U E U E E ble), 10), Closed Contour-4 ( 10, ∞, ∞(unstable)); the equilibrium state E10 from two branches coalesce and form in to a single branch with change in

α.

119 (a) α = 0.30 (b) α = 0.32

E U Figure 5.39: Enlarged pictures of G∗ as function of β, Closed Contour-3 ( 9 ,

E E E U E U E E 10, 10 (unstable), 10) , Closed Contour-4 ( 10, ∞, ∞ (unstable)); the equilibrium state E10 from two branches coalesce and form in to a single branch with change in α.

120 (a) α = 0.42 (b) α = 0.46

E U E Figure 5.40: Enlarged view of Closed Contour-2 ( 8 , 9,

E E U E U E E 9(unstable), 9 ), Closed Contour-3 ( 9 , 10, 10(unstable),

E U E E E U 10, ∞, ∞(unstable)), with increase in α the branches 9 on both the contours start merging.

121 (a) α = 0.58 (b) α = 0.62

E U E E Figure 5.41: Enlarged view of Closed Contour-1 ( 7 , 8, 8 (un-

E U E U E E E U stable), 8 ) and Closed Contour-2 ( 8 , 9, 9 (unstable), 9 ,

E E E U E E 10, 10 (unstable), 10, ∞, ∞ (unstable)), with increase in α

E U the branches 8 on both the contours start merging.

122 (a) α = 0.82 (b) α = 0.86

E U E U Figure 5.42: Enlarged view of Open Contour ( 6 , 7 ) and Closed Contour-1

E U E E E U E U E ( 7 , 8, 8 (unstable), 8 ) and Closed Contour-2 ( 8 , 9,

E E U E E E U E E 9(unstable), 9 , 10, 10 (unstable), 10, ∞, ∞ (unsta-

E U ble)), with increase in α the branches 7 on both the contours start merging.

123 (a) G∗ vs β (b) V,W vs β

Figure 5.43: (a) Global Coupling Force G∗, (b) Global Self-Consistency Variables

(V,W ), as a function of β, is now a single continuous single curve at α = 0.86.

124 5.4 Stability region

Though the global self-consistency constants in previous sections clearly indicate the stable and unstable equilibrium distributions at different α. Clear description of such stable and unstable distributions on an α , β plane is much more useful in knowing if the given distribution is stable or unstable.

E U The distributions i indicate when a distribution is unstable just due to the geometry (visibility) i.e. when the resonator is on unstable part of X(s) < 0 branch, then they are unstable as seen in Figure 5.5b (s7), here two eigenvalues are purely real and one of which is positive indicating this distribution is unstable. Similarly in each of the isolated/merged branches (either global self-consistency (V,W ) or in X(si) indicating the state of resonator in any equilibrium distributions) at a given value of

β, there exists two equilibrium distributions indicated with the same nomenclature, but one is stable and the other being unstable. For example at (α , β) = (0.2 , 4.5) the two equilibrium distributions are shown in Figure 5.44 both of which appear

E 1 E 2 similar geometrically but only 8 is stable and 8 is unstable(this 1 and 2 is used for making a distinction). This stability of this state was evaluated using the discrete spectrum Eq.( 3.32), the eigenvalues obtained in each of these distributions are shown in Figure 5.45, where in Figure 5.45a represents the eigenvalues of stable distribution where the real part is negative and Figure 5.45b represents eigenvalues of unstable distribution in which the real part of one eigenvalue is positive.

Even the equilibrium distributions similar to that is shown in Figure 5.44a

125 8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

E1 E2 (a) 8 (b) 8

Figure 5.44: Equilibrium distributions at (α , β) = (0.2 , 4.5), where they look similar

E 1 E 2 geometrically each named as 8 (stable), 8 (unstable).

5 5 ) ) 0 0 Im( Im(

-5 -5 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 Re( ) Re( )

E1 E2 (a) 8 (b) 8

E 1 E 2 Figure 5.45: Spectrum plots of distributions 8 (stable) shown in (a) and 8 (unstable) shown in (b), when (α , β) = (0.2 , 4.5).

126 12 10 8 6 4 B 2 A 0 0 0.5 1 1.5 2 2.5

Figure 5.46: A curve separating stable and unstable regions on (α , β) plane for equilibrium E7 distribution, A and B are two discrete points of (α , β) for which the distribution is stable and unstable respectively.

also would loose stability by Hopf criteria, i.e pair of eigenvalues with non-zero imaginary part and negative real part crosses to positive side. On a plane of coupling parameters (α , β) a line separating the stable and unstable parts of distributions in

Figure 5.46 for distribution E7 are shown here and A,B are two discrete points of

(α , β) for which the distribution is stable and unstable respectively.

Equilibrium distribution for the coupling parameters at the points A and

127 8 8

6 6

4 4 R(s) R(s)

2 2

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

(a) (α , β) = (1.6 , 1.8) (b) (α , β) = (1.9 , 3.0)

Figure 5.47: Equilibrium distributions at for two different coupling points at A when

(α , β) = (1.6 , 1.8) is shown in (a) and at B when (α , β) = (1.9 , 3.0) is shown in

(b).

B are shown in Figure 5.47. The spectrum plot for the parameters at A shown in Figure 5.48a have the real part of eigenvalues negative, hence for this coupling parameters the distribution is stable. Also for the coupling parameters mentioned at point B, the real part of two eigenvalues is positive as seen in Figure 5.48b, hence the above distribution for the coupling parameters at point B is unstable.

Just explained previously the stable and unstable regions on α , β plane for different equilibrium distributions E8 , E9 , E10 , E∞ is shown in Figure 5.49. All the

Figures 5.49a to 5.49c curves follow similar pattern, it can be clearly observed that these equilibrium states at higher α have lower β values for which they are stable. It is also observed that the area under the curves for distributions E8 , E9 , E10

128 10 10

5 5 ) ) 0 0 Im( Im( -5 -5

-10 -10 -0.4 -0.2 0 0.2 -0.4 -0.2 0 0.2 Re( ) Re( )

(a) (α , β) = (1.6 , 1.8) (b) (α , β) = (1.9 , 3.0)

E 1 Figure 5.48: Spectrum plot of distribution 8 (stable) is in Figure(a) and for distribu-

E 2 tion 8 is in (b), this corresponds two different coupling parameters A and B inside and outside of the stable region as shown in Figure 5.46.

129 keeps decreasing, clearly indicating that distributions in which more resonators lie on

X(s) < 0 have lesser stable region. The curve in Figure 5.49d representing equilibrium distribution E∞ is shown, here the curve doesn’t touch positive β axis, indicating this distribution is stable for large values of β for α ≤ 0.4962. It has to be noted that the curves in Figure 5.49 separates the regions in which these equilibriums exist, not the whole plane.

The equilibrium distribution E∞ at a higher values of β shown in Figure 5.50 are stable. The stability of these solutions can be seen from spectrum plots in Fig- ure 5.51, where the real part of eigenvalues are less than zero. Although for this range of α ([0.0 , 0.49]) the equilibrium distribution E∞ exists for β > 40.0 it has been found that this equilibrium still exists, but those values of coupling cannot be realized for real physical systems.

5.5 Summary

The value of coupling parameters α , β decides the number of possible equilibrium distributions, X(si) is a variable which is used to indicate the state of a resonator(its sign), any distribution for a given value of coupling parameterα , β can be uniquely determined by V,W along with X(si). Unlike monostable and linear systems the effect of α , β on non-linear systems cannot be decoupled and is not so straightforward.

From the analysis shown in this chapter, when α , β are largely negative,(i.e positive damping and negative stiffness), only an isolated monostable solution exists. However one can produce the Duffing like curve either by increase the external force F0 or non- 130 10 10

5 5

0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5

(a) E8 (b) E9

10 10

5 5

0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5

Figure 5.49: The curves separating stable and unstable regions on α , β plane for different equilibrium states E8 , E9 , E10 , and E∞ are shown here.

131 10 10

R(s) 5 R(s) 5

0 0 0.9 0.95 1 1.05 1.1 0.9 0.95 1 1.05 1.1 m(s) m(s)

(a) (α , β) = (0.45 , 20) (b) (α , β) = (0.45 , 40.0)

Figure 5.50: Equilibrium distribution E∞ at (a) β = 20, (b) β = 40, when α = 0.45.

2 2

1 1 ) ) 0 0 Im( Im( -1 -1

-2 -2 -2 -1 0 1 2 -2 -1 0 1 2 Re( ) Re( )

(a) (α , β) = (0.45 , 20.0) (b) (α , β) = (0.45 , 40.0)

Figure 5.51: Eigenvalues E∞ at (a) β = 20, (b) β = 40, when α = 0.45, indicates these equilibrium distributions are stable.

132 linear stiffness coefficient γ. In order to study the effect of β, on these systems α was held constant and β is varied, it is observed that multiple solutions starts appearing when β is increased. Two bifurcations that are common to a Duffing response at the upper and lower saddle-nodes are witnessed even in the case of coupled resonators, here they are referred to as Type-I and Type-II bifurcations. This phenomenon is explained with the help of a 3D-plot of (V , W , β) (named as contours) at a given α and also on a 2-D plot of (G∗ , β), where G∗ stands for global coupling force (net force acting on all the resonators in the population due to coupling and external force).

The isolated contours of (V , W , β) at lower values of α represents just two equilibrium distributions, increasing α which essentially pumps the energy in to the array of resonators, make the resonators on the lower branch to change to unstable branch, introducing a new equilibrium distribution on the same isolated contour.

Finally with more increase in α, each isolated contour start coming closer to each other and start merging, forming a single continuous contour. This indicates that the equilibrium distributions are not an isolated local phenomenon that occurs, but these distributions are dependent on each other for given set of parameters α , β.

The stability of distributions is found by evaluating the eigenvalues, from discrete spectrum equation. On α , β plane the boundaries separating the unstable regions for each distribution has also been obtained.

133 CHAPTER VI

CONCLUSION AND FUTURE WORK

6.1 Summary

In this dissertation, the modeling of large array non-linear coupled resonators has been discussed. In particular, the continuum approach of modeling both discrete system of N resonators and extending it to continuous system has been formulated here. The coefficient of non-linearity (γ) along with global coupling coefficients (α , β) influences the response of these coupled resonators, they in toto decide ifthereisan unique monostable response or if multiple states exist for the system of resonators.

Microdevices offer several advantages in terms of power consumption, sensitivity and cost compared to their macroscale counterparts. MEMS devices were commercially utilized in the making of ink-jet print heads, accelerometers, gyroscopes, pressure sensors, and magnetometers. Microresonators are components that respond at the well-defined frequency when actuated by an external excitation, single resonator MEMS devices are used in chemical, inertial and acoustic sensing, atomic force based microscopy, computing, and radio-frequency communications, but they suffer from severe throughput limitations i.e., the amount of information theycan sense/process. This limitation was initially overcome by exploiting massive paral-

134 lelization of large arrays uncoupled resonators. Recent studies suggested that constraints on the throughput can be addressed by exploiting the collective emergent complex behaviors like synchronization, localization, spatial confinement by coupling those arrays of coupled micro/nanoresonators. This document is an attempt to develop analysis methods to study the dynamics of linear and nonlinear dynamics of large arrays of coupled resonators operating under the influence of parametric uncertainty.

To study a mutually-coupled system, even of the relatively manageable second- order oscillators that dominate the literature of physics and electrical engineering,

2N order non-linear differential equation, enough research has been done from the time of Vanderpol oscillators. A.T.Winfree in his research have modeled each of this biological subsystems as oscillators differing slightly in their frequencies and are self- sustaining to collective rhythm due to the coupling present in the population, this is analogous to a mean-filed theory. Strogatz and Mirollo extended these models to continuum and eventually studied on the stability, unfortunately, these methods are ill-suited for many practical problems, which feature both amplitude and phase dynamics, inherent element-level parameter variation (mistuning), nonlinearity, and/or noise. So a coupled microresonator array with reactive and/or dissipative coupling, together with element-level dynamics that contain stiffness nonlinearities under the influence of external excitation is taken up for developing the new analytical framework.

The system considered in this dissertation consists of an electromagnetically

135 transduced-microresonator array, actuated by the Lorentz force, which is generated by the interaction between an external permanent magnet and the integrated current loop on the resonator. Experimental results indicate that the parametrically excited systems exhibit moderately large amplitudes near resonance and modeling must allow large elastic displacements, making it a good case to include cubic nonlinearity in to the model. The method attempted is to reduce the coupled differential equations of resonators to an integro-differential equation which can represent both finite and infinite models using an indexing variable si. The obtained equations are taken care of all the possible parameter variations which include that of coupling, can be solved by applying standard analytical techniques of non-linear dynamics. Here the method of multiple time scales has been used to obtain the slow-flow equations, that describes dynamics of amplitude components. Equilibrium distributions are non-linear integral equations, this can be iteratively solved for global-self consistency variables

V,W (they are related to amplitude components of population of resonators) by minimizing the fmincon error function in MATLAB. The stability about equilibrium distributions can be determined by solving for the eigenvalues of linearized slow-flow equations, which essentially gives the discrete spectrum.

It has to be noted that the non-linear integral equations may have multiple equilibrium states, depending on the non-linear stiffness coefficient γ and coupling parameters. That happens only when γ is sufficiently large, but when γ ≈ 0 or relatively small, the solution is always unique also called mono-stable equilibrium distribution. In case of uniformly distributed detuned-resonator the effect of

136 coupling parameters(α , β) on the response characteristics of populations are analyzed. As expected the resonators which have slightly detuned frequencies with respect to actuating forcing frequency have higher amplitude response and exhibit least phase(lag/lead) compared to the resonators that have higher detuned frequency, the response here is similar to Lorentzian distribution. α here stands for negative damping, by increasing it would pump energy in to the system and leads to increase in overall amplitude of system. β here stands for negative stiffness of the population, increasing this shifts collective lag(lead) of the population and also decreases the overall response of population. Global self-consistency variables V,W as a function for the population of uniform distributions N discrete resonators is effected by their number of resonator N(even or odd). The spread in (V,W ) is more significant for N = 20 as compared to N = 500 indicating that the stationary response is more sensitive to the population distribution for fewer numbers of resonators.

The value of coupling parameters α , β decides the number of possible equilibrium distributions, X(si) is a variable which is used to indicate the state of a resonator(its sign), any distribution for a given value of coupling parameterα , β can be uniquely determined by V,W along with X(si). Unlike monostable and linear systems the effect of α , β on non-linear systems cannot be decoupled and is not so straightforward. From the analysis shown in this chapter, when α , β are largely negative,(i.e positive damping and negative stiffness), only an isolated monostable solution exists. However one can produce the Duffing like curve either by increase the external force F0 or non-linear stiffness coefficient γ. In order to study the effect

137 of β, on these systems α was held constant and β is varied, it is observed that multiple solutions starts appearing when β is increased. Two bifurcations that are common to a Duffing response at the upper and lower saddle-nodes are witnessed even inthecase of coupled resonators, here they are referred to as Type-I and Type-II bifurcations.

This phenomenon is explained with the help of a 3D-plot of (V , W , β) (named as contours) at a given α and also on a 2-D plot of (G∗ , β), where G∗ stands for global coupling force (net force acting on all the resonators in the population due to coupling and external force).

The stability of distributions is found by evaluating the eigenvalues, from discrete spectrum equation. On an α , β the boundaries separating the unstable regions for each distribution has also been obtained.

138 6.2 Suggested Future work

The current work focused on the non-linearity in stiffness and linear terms of coupling, the scope of this research can be expanded by including the terms which included nonlinear terms of coupling parameters and damping and understanding the parametric variations of such models. It would be also a nice attempt to include higher-order non-linear stiffness terms in order to make models closer to the real physical systems.

The problem taken here for consideration is under the influence of a pure harmonic excitation, the modeling techniques developed here can also be extended to any generalized excitations that are not only includes different types of loads but which are applied at varying degree across the population.

The coupling involved in this problem is uniform across all the resonators in the population, but the future research may require an approach where localized variation in the coupling is taken care, so that the localized parametric variation on the global behavior of array can be understood.

The stability of various equilibrium distributions discussed here has taken

Hopf criteria, however the limit cycle behavior of large array resonators could not be taken, behavior of global self-consistency variables at such values of coupling could be area of interest.

Correlating the behavior of mathematical models with potential experimental models may lead to development of better mathematical models, which further enhances our understanding of such systems.

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