DOCSLIB.ORG

Some Dynamical Properties of the Family of Tent Maps*

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Some Dynamical Properties of the Family of Tent Maps* Int. Journal of Math. Analysis, Vol. 7, 2013, no. 29, 1433 - 1449 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2013.3361 Some Dynamical Properties of the Family of Tent Maps* Wadia Faid Hassan Al-shameri 1 and Mohammed Abdulkawi Mahiub 2 1 Department of Mathematics, Faculty of Science and Arts, Najran University [email protected] 2 Department of Mathematics, Faculty of Science and Arts, Najran University [email protected] Copyright © 2013 Wadia Faid Hassan Al-shameri and Mohammed Abdulkawi Mahiub. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper, we investigated some dynamical behavior of family of tent maps. Such dynamical properties include fixed points and their stability, period orbits, visualize the iterations using a kind of plot called a cobweb plot and demonstrate bifurcation diagram forTr . Furthermore, detecting the presence of chaos in the discrete dynamical systemTr is investigated. Finally, we develop Matlab computer programs that reflect the results interpreting such dynamical behavior. Mathematics Subject Classification: 37C25, 37E15, 34C23 Keywords: Fixed points, periodic points, periodic orbits, bifurcation 1. INTRODUCTION Consider the parameterized tent map, which can be described piecewise by ⎧2,ifrx 0≤≤ x 1 Tx()= 2 r ⎨ 1 ⎩2(1rx− ),if 2 <≤ x1 1434 Wadia Faid Hassan Al-shameri and Mohammed Abdulkawi Mahiub where 02<≤r . This map is continuous, linear on each of the intervals (−1, 1 / 2] and [1/2, 1) (with respect to slopes r and −r ) and has the points (0,0) and (1, 0) in its graph. This map is known as tent maps and often introduced as one of the first examples of chaotic maps literature for nonlinear discrete dynamical systems. Its dynamics exhibit various features are commonly used to identify chaotic systems (see for instance [5]). Chaos theory describes complex motion and the dynamics of sensitive systems. Chaotic systems are mathematically deterministic but nearly impossible to be predicted. Chaos is more evident in long-term systems than in short-term systems. Behavior in chaotic systems is a periodic, in other words, no variable describes the state of the system that undergoes a regular repetition of values. A chaotic system can actually evolve in a way that appears to be smooth and ordered; however, chaos refers to the issue of whether or not it is possible to make accurate long-term predictions of any system if the initial conditions are known to an accurate degree [8]. In addition to the sensitivity condition, there is a mixing condition called transitivity and a regularity condition called density of periodic points[2]. Deterministic chaos can be described by deterministic map, such as tent map Tr as follows: starting from any point between 0 and 1, the trace of the map would still lie in the range [0, 1] of the tent map, which is a absorbing set. Also, each point inside [0, 1] will be often visited arbitrarily closely and infinitely by any ’typical’ solution, hence it is also an attractor [2]. The purpose of this paper is to investigate the dynamical behavior of the family of tent maps, we will limit ourselves to the case 02< r ≤ [3]. For certain parameter values, the mapping undergoes stretching and folding transformations and displays sensitivity to initial conditions and periodicity (see [1] and [6]). The tent map is studied in the mathematics of dynamical systems. Because of its simple shape, the tent map shape under iteration is very well understood. And despite its simple shape, it has several interesting properties [10]. The paper is organized as follows; The first section is the introduction and the second section describes the dynamical behavior for this discrete dynamical systemTr such as fixed points and their stability and period orbits followed by section three, which presents cobweb plots. Section four demonstrates period doubling and obtains the bifurcation diagram ofTr , for the parameter 02.<≤r The dramatic bifurcation that occurs at r = 1 will be explained. The results will be obtained using Matlab computer programs. One of these programs is to visualize the iterations using a kind of plot called a cobweb plot. Sketching the bifurcation diagram ofTr , for the parameter 01< r ≤ is by using another Matlab program. Fixed points will show up as a single point, a periodic orbit as several points, and a chaotic orbit as a band or several bands of points. Section five presents the results that are detecting the presence of chaos in the tent mapsTr and examining sensitivity to initial conditions; it is one of the dynamical Some dynamical properties of the family of tent maps 1435 properties of strange attractors. Results and discussions will be displayed. Finally, in the last section some concluding remarks will be highlighted. 2. FIXED POINTS AND PERIODIC ORBITS Consider an iteration x = fx() (2.1) nn+1 A fixed point, or point of period one, of discrete dynamical system (2.1) is a point at which xnnn+1 ==fx() x for all n[7] For the tent map, this implies thatTxrn()= x n, for all n. Graphically, the fixed points can be found by identifying intersections of the map Txr ()with the line y = x , see [3]. Figures 2.1(a) through 2.1(c) displayTr for r = 2/7,1/2 and 5/6by Matlab program 1 in the Appendix. As r increases, the height of the graph ofTr rises, because of the factor r in the formula forTr . From this observation and the three graphs in Figure 2.1; we deduce that if 01/2<<r , thenTr intersects the line y = x once (at 0), whereas if 1/2<<r 1, then there are two points of intersection. We are led to analyze separately the members of{ Tr } for which 01/2<<r , r = 1/2 and 1/2< r < 1. Finally, we will studyT1 , which is the original tent mapT and which has some very interesting features, we shall follow [3]. Graphical iteration of tent map:T (x) r Graphical iteration of tent map:T (x) 1 r 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 r =0.5. r =0.2857. r 0.4 (x), is in (0, 2] r 0.4 (x), is in (0, 2] r T 0.3 T 0.3 0.2 0.2 0.1 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x x Figure 2.1 (a) Figure 2.1 (b) 1436 Wadia Faid Hassan Al-shameri and Mohammed Abdulkawi Mahiub Graphical iteration of tent map:T (x) r 1 0.9 0.8 0.7 0.6 0.5 r =0.8333. r (x), isin (0, 2] 0.4 r T 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Figure 2.1(c) Case 1. 0<<r 1/ 2 The graph in Figure 2.1(a) shows that 0 is the only fixed point of Tr . Since 01/2<<r , it follows from the definition of Tr that if 0 < x ≤ 1/ 2 , then 02≤ Txr ( ) =< rxx and if 1/ 2 < x ≤ 1, then −1 02(1)1≤=−<−<<Tx() r x x x r 2 ∞ Consequently for any x ∈[0,1], the sequence T{n} x is bounded and decreasing and by { μ ()}n=0 the continuity of Tr , the sequence converges to the fixed point 0. Therefore, 0 is an attracting fixed point whose basin of attraction is [0,1] . Case 2. r = 1/2 First we notice that if 0 ≤ x ≤ 1/ 2 , then T1/ 2 (x) = 2(1/ 2)x = x , so that x is a fixed point of T1/ 2 (Figure 2.1(b)). Next, we calculate that 1/ 2 < x ≤ 1, then 1 02(1/2)(1)1≤=Tx() −=−≤ x x r 2 So that Txx1/2 ()= is a fixed point of T1/ 2 . Consequently, every point in [0,1] either is a fixed point of T1/ 2 or has a fixed point its first iterate. Case 3. 1/2<<r 1 In addition to the fixed point 0, there is a second fixed point p that lies in (1 / 2,1] , as one can see in Figure 2.1(c). To evaluate p we solve the equation p = Tpr ( ) =−2(1 r p ) which yields Some dynamical properties of the family of tent maps 1437 2r p = 12+ r / As r increases from ½ toward 1, p increases from 1/2 toward 2/3. Because Txr ()=>21 r on [0,1] except at 1/2, both 0 and p are repelling fixed points. The period-2 points of Tr [2] are the fixed points of Tr , which is given by ⎧401/4rx2 for≤≤ x r ⎪ [2] ⎪2(12rrxforrx−<≤ ) 1/4 1/2 Txr ()= ⎨ ⎪2rrrxforx (1−+ 2 2 ) 1/2 <≤− 1 1/4 r ⎪ 2 ⎩4(1)rxforrx−−≤≤ 11/4 1 [2] The graph of T8 appears in Figure 2.2 plotted by Matlab program 2 in the Appendix, [2] and suggests that for 1/2<<r 1, Tr has four fixed points that can be found by solving [2] [2] the four equations x = Txr () arising from the definition of Tr . We find that the fixed 22rr 4 r2 points are0, , , and 14++rr22 12 14 + r Graphical iteration of tent map, r=0.8333 1 0.9 0.8 0.7 0.6 0.5 (x), n=2 (x), [n] T 0.4 0.3 0.2 0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x Figure 2.2 ⎧ 24rr2 ⎫ The first and third are the two fixed points of Tr , so it follows that ⎨ 22, ⎬ ⎩⎭14++rr 14 [2]′ 2 is a 2-cycle forTr .
Load more

Recommended publications
  • Construction and Chaos Properties Analysis of a Quadratic Polynomial Surjective Map
    INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 11, 2017 Construction and Chaos Properties Analysis of a Quadratic Polynomial Surjective Map Jun Tang, Jianghong Shi, Zhenmiao Deng randomly distributed sequences [16]–[18]. Abstract—In this paper, a kind of quadratic polynomial surjective In this paper, we deduce the relationships between the map (QPSM) is constructed, and the topological conjugation of the coefficients of a quadratic polynomial surjective map (QPSM) QPSM and tent map is proven. With the probability density function and prove the topological conjugacy of the QPSM and tent map. (PDF) of the QPSM being deduced, an anti-trigonometric transform We give the PDF of the QPSM, which is then used to design a function is proposed to homogenize the QPSM. The information entropy, Kolmogorov entropy (KE), and discrete entropy (DE) of the transform function to homogenize the QPSM sequence. Finally, QPSM are calculated for both the original and homogenized maps we estimate entropies of the QPSM such as the information, with respect to different parameters. Simulation results show that the Kolmogorov, and discrete entropies for both the original and information entropy of the homogenized sequence is close to the homogenized map. theoretical limit and the discrete entropy remains unchanged, which This paper is organized as follows. Section II presents a suggest that the homogenization method is effective. Thus, the method for determining the QPSM coefficients, and the homogenized map not only inherits the diverse properties of the original QPSM but also possesses better uniformity. These features topological conjugation of the QPSM and tent map is proven. make it more suitable to secure communication and noise radar.
    [Show full text]
  • Topological Entropy of Quadratic Polynomials and Sections of the Mandelbrot Set
    Topological entropy of quadratic polynomials and sections of the Mandelbrot set Giulio Tiozzo Harvard University Warwick, April 2012 2. External rays 3. Main theorem 4. Ideas of proof (maybe) 5. Complex version Summary 1. Topological entropy 3. Main theorem 4. Ideas of proof (maybe) 5. Complex version Summary 1. Topological entropy 2. External rays 4. Ideas of proof (maybe) 5. Complex version Summary 1. Topological entropy 2. External rays 3. Main theorem 5. Complex version Summary 1. Topological entropy 2. External rays 3. Main theorem 4. Ideas of proof (maybe) Summary 1. Topological entropy 2. External rays 3. Main theorem 4. Ideas of proof (maybe) 5. Complex version Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous.
    [Show full text]
  • PERIODIC ORBITS of PIECEWISE MONOTONE MAPS a Dissertation
    PERIODIC ORBITS OF PIECEWISE MONOTONE MAPS A Dissertation Submitted to the Faculty of Purdue University by David Cosper In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 2018 Purdue University Indianapolis, Indiana ii ACKNOWLEDGMENTS I would like to thank my advisor Micha l Misiurewicz for sticking with me all these years. iii TABLE OF CONTENTS Page LIST OF FIGURES ::::::::::::::::::::::::::::::: iv ABSTRACT ::::::::::::::::::::::::::::::::::: v 1 INTRODUCTION :::::::::::::::::::::::::::::: 1 2 PRELIMINARIES :::::::::::::::::::::::::::::: 4 2.1 Kneading Theory :::::::::::::::::::::::::::: 4 2.2 Sharkovsky's Theorem ::::::::::::::::::::::::: 12 2.3 Periodic Orbits of Interval Maps :::::::::::::::::::: 16 3 PERIODIC ORBITS OF PIECEWISE MONOTONE MAPS ::::::: 20 3.1 Horizontal/Vertical Families and Order Type ::::::::::::: 23 3.2 Extremal Points ::::::::::::::::::::::::::::: 35 3.3 Characterizing P :::::::::::::::::::::::::::: 46 4 MATCHING :::::::::::::::::::::::::::::::::: 55 4.1 Matching :::::::::::::::::::::::::::::::: 58 4.2 Topological entropy ::::::::::::::::::::::::::: 62 4.3 Transitivity ::::::::::::::::::::::::::::::: 64 4.4 Beyond transitivity ::::::::::::::::::::::::::: 67 LIST OF REFERENCES :::::::::::::::::::::::::::: 71 VITA ::::::::::::::::::::::::::::::::::::::: 73 iv LIST OF FIGURES Figure Page 1.1 An example of a two-sided truncated tent map. ::::::::::::: 2 2.1 This truncated tent map cannot have the min-max periodic orbit (RLR)1, and hence does not have a period 3 periodic point. ::::::::::: 15 2.2 The period 6 orbit with itinerary [R ∗ (RLR)]1. Notice that the first return map on the blue interval is f 2 and that the three points form a periodic orbit under f 2. :::::::::::::::::::::::::: 19 3.1 This truncated tent map cannot have the orbit (RLR)1. However, it does have the orbit (RLL)1. :::::::::::::::::::::::::: 21 3.2 This truncated tent map cannot have the orbit (RL)1.
    [Show full text]
  • Math 2030 Fall 2012
    Math 2030 Fall 2012 3 Definitions, Applications, Examples, Numerics 3.1 Basic Set-Up Discrete means “not continuous” or “individually disconnected,” dynamical refers to “change over time,” and system is the thing that we are talking about. Thus this course is about studying how elements in some ambient space move according to discrete time steps. The introductory material in this course will involve so-called one-dimensional systems. Nonetheless, we introduce the general notion of a Discrete Dynamical System since it will play a major role later. A Discrete Dynamical System consists of a set X (called the state space) and a map F ( ) (called the dynamic map) that takes elements in X to elements in X; that is, F ( ) : X· X. Given a point x X, the first iterate of x is defined by · → 0 ∈ 0 x1 = F (x0). The second iterate of x0 is nothing but the first iterate of x1, and is denoted 2 x2 = F (x1)= F F (x0) =: F (x0). Note that the power of F in the last expression is not a “power” in the usual sense, but the number of times F has been used in the iteration. The th process repeats, where the k iterate xk is given by xk = F (xk−1) = F F (xk−2) . = . = F F F(x ) ··· 0 k times = F k(x ) | {z0 } The orbit of the seed x is denoted by x is defined as the (ordered) collection of all of the 0 h 0i iterates of x0. Notationally, this means x = x ,x ,x ,... h 0i { 0 1 2 } = F k(x ): k = 0, 1, 2,..
    [Show full text]
  • C*-ALGEBRAS ASSOCIATED with INTERVAL MAPS 1. Introduction
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 359, Number 4, April 2007, Pages 1889–1924 S 0002-9947(06)04112-2 Article electronically published on November 22, 2006 C*-ALGEBRAS ASSOCIATED WITH INTERVAL MAPS VALENTIN DEACONU AND FRED SHULTZ Abstract. For each piecewise monotonic map τ of [0, 1], we associate a pair of C*-algebras Fτ and Oτ and calculate their K-groups. The algebra Fτ is an AI-algebra. We characterize when Fτ and Oτ aresimple.Inthosecases,Fτ has a unique trace, and Oτ is purely infinite with a unique KMS state. In the case that τ is Markov, these algebras include the Cuntz-Krieger algebras OA, and the associated AF-algebras FA. Other examples for which the K-groups are computed include tent maps, quadratic maps, multimodal maps, inter- val exchange maps, and β-transformations. For the case of interval exchange maps and of β-transformations, the C*-algebra Oτ coincides with the algebras defined by Putnam and Katayama-Matsumoto-Watatani, respectively. 1. Introduction Motivation. There has been a fruitful interplay between C*-algebras and dynam- ical systems, which can be summarized by the diagram dynamical system → C*-algebra → K-theoretic invariants. A classic example is the pair of C*-algebras FA, OA that Krieger and Cuntz [10] associated with a shift of finite type. They computed the K-groups of these alge- bras, and observed that K0(FA) was the same as a dimension group that Krieger had earlier defined in dynamical terms [34]. Krieger [35] had shown that this di- mension group, with an associated automorphism, determined the matrix A up to shift equivalence.
    [Show full text]
  • On the Topological Conjugacy Problem for Interval Maps
    On the topological conjugacy problem for interval maps Roberto Venegeroles1, ∗ 1Centro de Matem´atica, Computa¸c~aoe Cogni¸c~ao,UFABC, 09210-170, Santo Andr´e,SP, Brazil (Dated: November 6, 2018) We propose an inverse approach for dealing with interval maps based on the manner whereby their branches are related (folding property), instead of addressing the map equations as a whole. As a main result, we provide a symmetry-breaking framework for determining topological conjugacy of interval maps, a well-known open problem in ergodic theory. Implications thereof for the spectrum and eigenfunctions of the Perron-Frobenius operator are also discussed. PACS numbers: 05.45.Ac, 02.30.Tb, 02.30.Zz Introduction - From the ergodic point of view, a mea- there exists a homeomorphism ! such that sure describing a chaotic map T is rightly one that, after S ! = ! T: (2) iterate a randomly chosen initial point, the iterates will ◦ ◦ be distributed according to this measure almost surely. Of course, the problem refers to the task of finding !, if This means that such a measure µ, called invariant, is it exists, when T and S are given beforehand. −1 preserved under application of T , i.e., µ[T (A)] = µ(A) Fundamental aspects of chaotic dynamics are pre- for any measurable subset A of the phase space [1, 2]. served under conjugacy. A map is chaotic if is topo- Physical measures are absolutely continuous with respect logical transitive and has sensitive dependence on initial to the Lebesgue measure, i.e., dµ(x) = ρ(x)dx, being the conditions. Both properties are preserved under conju- invariant density ρ(x) an eigenfunction of the Perron- gacy, then T is chaotic if and only if S is also.
    [Show full text]
  • Improved Tent Map and Its Applications in Image Encryption
    International Journal of Security and Its Applications Vol.9, No.1 (2015), pp.25-34 http://dx.doi.org/10.14257/ijsia.2015.9.1.03 Improved Tent Map and Its Applications in Image Encryption Xuefeng Liao Oujiang College, Wenzhou University, Wenzhou, China 325035 [email protected] Abstract In this paper, an improved tent map chaotic system with better chaotic performance is introduced. The improved tent map has stronger randomness, larger Lyapunov exponents and C0 complexity values compare with the tent map. Then by using the improved tent map, a new image encryption scheme is proposed. Based on all analysis and experimental results, it can be concluded that, this scheme is efficient, practicable and reliable, with high potential to be adopted for network security and secure communications. Although the improved tent chaotic maps presented in this paper aims at image encryption, it is not just limited to this area and can be widely applied in other information security fields. Keywords: Improved tent map; Chaos performance; Image encryption; Shuffle; Diffuse 1. Introduction With the rapid development in digital image processing and network communication, information security has become an increasingly serious issue. All the sensitive data should be encrypted before transmission to avoid eavesdropping. However, bulk data size and high redundancy among the raw pixels of a digital image make the traditional encryption algorithms, such as DES, IDEA, AES, not able to be operated efficiently. Therefore, designing specialized encryption algorithms for digital images has attracted much research effort. Because chaotic systems/maps have some intrinsic properties, such as ergodicity, sensitive to the initial condition and control parameters, which are analogous to the confusion and diffusion properties specified by Shannon [1].
    [Show full text]
  • On the Beta Transformation
    On the Beta Transformation Linas Vepstas December 2017 (Updated Feb 2018 and Dec 2018) [email protected] doi:10.13140/RG.2.2.17132.26248 Abstract The beta transformation is the iterated map bx mod 1. The special case of b = 2 is known as the Bernoulli map, and is exactly solvable. The Bernoulli map provides a model for pure, unrestrained chaotic (ergodic) behavior: it is the full invariant shift on the Cantor space f0;1gw . The Cantor space consists of infinite strings of binary digits; it is notable for many properties, including that it can represent the real number line. The beta transformation defines a subshift: iterated on the unit interval, it sin- gles out a subspace of the Cantor space that is invariant under the action of the left-shift operator. That is, lopping off one bit at a time gives back the same sub- space. The beta transform seems to capture something basic about the multiplication of two real numbers: b and x. It offers insight into the nature of multiplication. Iterating on multiplication, one would get b nx – that is, exponentiation; the mod 1 of the beta transform contorts this in strange ways. Analyzing the beta transform is difficult. The work presented here is more-or- less a research diary: a pastiche of observations and some shallow insights. One is that chaos seems to be rooted in how the carry bit behaves during multiplication. Another is that one can surgically insert “islands of stability” into chaotic (ergodic) systems, and have some fair amount of control over how those islands of stability behave.
    [Show full text]
  • Private Digital Communications Using Fully-Integrated Discrete-Time Synchronized Hyper Chaotic Maps
    PRIVATE DIGITAL COMMUNICATIONS USING FULLY-INTEGRATED DISCRETE-TIME SYNCHRONIZED HYPER CHAOTIC MAPS by BOSHAN GU Submitted in partial fulfillment of the requirements For the degree of Master of Science Thesis Adviser: Dr. Soumyajit Mandal Department of Electrical, Computer, and Systems Engineering CASE WESTERN RESERVE UNIVERSITY May, 2020 Private Digital Communications Using Fully-Integrated Discrete-time Synchronized Hyper Chaotic Maps Case Western Reserve University Case School of Graduate Studies We hereby approve the thesis1 of BOSHAN GU for the degree of Master of Science Dr. Soumyajit Mandal 4/10/2020 Committee Chair, Adviser Date Dr. Christos Papachristou 4/10/2020 Committee Member Date Dr. Francis Merat 4/10/2020 Committee Member Date 1We certify that written approval has been obtained for any proprietary material contained therein. Table of Contents List of Tables v List of Figures vi Acknowledgements ix Acknowledgements ix Abstract x Abstract x Chapter 1. Introduction1 Nonlinear Dynamical Systems and Chaos1 Digital Generation of Chaotic Masking7 Motivation for This Project 11 Chapter 2. Mathematical Analysis 13 System Characteristic 13 Synchronization 16 Chapter 3. Circuit Design 19 Fully Differential Operational Amplifier 19 Expand and Folding function 27 Sample and Hold Circuits 31 Digital Signal Processing Block and Clock Generator 33 Bidirectional Hyperchaotic Encryption System 33 Chapter 4. Simulation Results 37 iii Chapter 5. Suggested Future Research 41 Current-Mode Chaotic Generators for Low-Power Applications 41 Appendix. Complete References 43 iv List of Tables 1.1 Different chaotic map and its system function.6 1.2 Similar factors between chaotic encryption and traditional cryptography.8 3.1 Op-amp performance simulated using different process parameter 23 3.2 Different gate voltage of transistor in corner case simulation 23 4.1 System performance comparison between previous work and this project.
    [Show full text]
  • A Study of Chaotic Maps Producing Symmetric Distributions in the Fish School Search Optimization Algorithm with Exponential Step Decay
    S S symmetry Article A Study of Chaotic Maps Producing Symmetric Distributions in the Fish School Search Optimization Algorithm with Exponential Step Decay Liliya A. Demidova * and Artyom V. Gorchakov * Institute of Integrated Safety and Special Instrumentation, Federal State Budget Educational Institution of Higher Education «MIREA–Russian Technological University», 78, Vernadskogo Avenye, 119454 Moscow, Russia * Correspondence: [email protected] (L.A.D.); [email protected] (A.V.G.) Received: 17 April 2020; Accepted: 6 May 2020; Published: 8 May 2020 Abstract: Inspired by the collective behavior of fish schools, the fish school search (FSS) algorithm is a technique for finding globally optimal solutions. The algorithm is characterized by its simplicity and high performance; FSS is computationally inexpensive, compared to other evolution-inspired algorithms. However, the premature convergence problem is inherent to FSS, especially in the optimization of functions that are in very-high-dimensional spaces and have plenty of local minima or maxima. The accuracy of the obtained solution highly depends on the initial distribution of agents in the search space and on the predefined initial individual and collective-volitive movement step sizes. In this paper, we provide a study of different chaotic maps with symmetric distributions, used as pseudorandom number generators (PRNGs) in FSS. In addition, we incorporate exponential step decay in order to improve the accuracy of the solutions produced by the algorithm. The obtained results of the conducted numerical experiments show that the use of chaotic maps instead of other commonly used high-quality PRNGs can speed up the algorithm, and the incorporated exponential step decay can improve the accuracy of the obtained solution.
    [Show full text]
  • Mostly Conjugate: Relating Dynamical Systems — Beyond Homeomorphism
    Mostly Conjugate: Relating Dynamical Systems — Beyond Homeomorphism Joseph D. Skufca1, ∗ and Erik M. Bollt1, † 1 Department of Mathematics, Clarkson University, Potsdam, New York (Dated: April 24, 2006) A centerpiece of Dynamical Systems is comparison by an equivalence relationship called topolog- ical conjugacy. Current state of the field is that, generally, there is no easy way to determine if two systems are conjugate or to explicitly find the conjugacy between systems that are known to be equivalent. We present a new and highly generalizable method to produce conjugacy functions based on a functional fixed point iteration scheme. In specific cases, we prove that although the conjugacy function is strictly increasing, it is a.e. differentiable, with derivative 0 everywhere that it exists — a Lebesgue singular function. When applied to non-conjugate dynamical systems, we show that the fixed point iteration scheme still has a limit point, which is a function we now call a “commuter” — a non-homeomorphic change of coordinates translating between dissimilar systems. This translation is natural to the concepts of dynamical systems in that it matches the systems within the language of their orbit structures. We introduce methods to compare nonequivalent sys- tems by quantifying how much the commuter functions fails to be a homeomorphism, an approach that gives more respect to the dynamics than the traditional comparisons based on normed linear spaces, such as L2. Our discussion includes fundamental issues as a principled understanding of the degree to which a “toy model” might be representative of a more complicated system, an important concept to clarify since it is often used loosely throughout science.
    [Show full text]
  • Dynamics of Unimodal Interval Maps Notes for the Mini-Course at the University of Vienna, May 2017
    Dynamics of unimodal interval maps Notes for the mini-course at the University of Vienna, May 2017 Ana Anuˇsi´c,University of Zagreb Version of May 4, 2017 1 Discrete dynamical systems - basic definitions Definition 1.1. Dynamical system is a pair (X; f), where X is a compact metric space and f : X ! X is (piecewise) continuous. For n 2 N denote by f n := f ◦f ◦:::◦f (n times). The forward orbit of x 2 X is Orb(x) := fx; f(x); f 2(x); f 3(x);:::g: The !-limit set of x is a set of accumulation points of Orb(x), i.e., ni !(x; f) = fy 2 X : there exists a strictly increasing (ni)i2N; f (x) ! y as i ! 1g: We say that x is periodic if there exists n 2 N such that f n(x) = x. The smallest such n 2 N is called a prime period of x. If x has period one, it is called a fixed point. If x is not periodic but there exists m 2 N such that f m(x) is periodic, then x is called preperiodic. If x 2 !(x), then x is called recurrent. Definition 1.2. Let (X; f) and (Y; g) be dynamical systems. We say that the systems (or maps f, g) are topologically conjugate if there exists a homeomorphism h: X ! Y such that h ◦ f = g ◦ h. Such h is called a topological conjugation of f and g. If h can be taken at most continuous and surjective, systems are topologically semi-conjugate and g is called a factor of f.
    [Show full text]