DOCSLIB.ORG

A Dynamical Systems Approach to Modeling Input-Output Systems Martin Casdagli Santa Fe Institute, 1120 Canyon Road Santa Fe, New Mexico 87501

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Dynamical Systems Approach to Modeling Input-Output Systems Martin Casdagli Santa Fe Institute, 1120 Canyon Road Santa Fe, New Mexico 87501 A Dynamical Systems Approach to Modeling Input-Output Systems Martin Casdagli Santa Fe Institute, 1120 Canyon Road Santa Fe, New Mexico 87501 Abstract Motivated by practical applications, we generalize theoretical results on the nonlinear modeling of autonomous dynamical systems to input-output systems. The inputs driving the system are assumed to be observed, as well as the out­ puts. The underlying dynamics coupling inputs to outputs is assumed to be deterministic. We give a definition of chaos for input-output systems, and develop a theoretical framework for state space reconstruction and modeling. Most of the results for autonomous deterministic systems are found to gener­ alize to input-output systems with some modifications, even if the inputs are stochastic. 1 Introduction There has been much recent interest in the nonlinear modeling and prediction of time series data, as evidenced by this conference proceedings volume. For approaches mo­ tivated by deterministic chaos, see [2, 5, 7, 8] and references therein. For approaches motivated by nonlinear stochastic models, see [15] and references therein. In both these approaches, it is assumed that a time series x(t) is obtained from observations of an autonomous dynamical system, possibly perturbed by unobserved forces or noise. By contrast, when modeling input-output systems, in addition to an observed output time series x(t), there is also available an observed input time series u(t); see Figure 1. Modeling input-output systems is appropriate in many applications. For example, in vibration testing, a mechanical device may be subjeeted to a controlled random forcing to test its robustness in a simulated environment. Also, in scientific experi­ ments, one may be interested in the response of a system to various forms of stimulus. There are many other disciplines, for example meteorology and economics, in which pairs of input-output time series from input-output systems are available for analysis. In this paper we will develop a theory for the nonlinear modeling of input-output systems based on deterministic dynamics. This deterministic approach assumes that the input-output time series arises from a finite dimensional dynamical system ds dt = f(s{t),u(t)) (1) x(t) = h(s(t)) (2) 1 Input u(t) Unknown Output x(t) System Figure 1: Conceptual model of a single input-single output system where s(t) E ~d denotes a d-dimensional state, and for simplicity we take u(t) E ~ to be a scalar input, f : ~d X ~ --> ~d to be a smooth flow, and h : ~d --> ~ to be a scalar measurement function. It is then natural to attempt to model and forecast the behavior of the input-output system with a nonlinear deterministic model of the form x(t) = P(x(t - r), x(t - 2r), .. , x(t - mr), u(t), u(t - r), .. ,u(t - (1- l)r)), (3) where P is a nonlinear function fitted to the input-output time series data. This deterministic approach has been applied by Hunter to a variety of practical examples [11]. Of course, the deterministic system (1,2) is only an approximation to reality. Firstly, it is assumed that only d independent modes of the system are excited to a significant amplitude, where in practice d is reasonably small. Secondly, it is assumed that effects due to unobserved sources of noise and measurement errors are small enol.\gh to be ignored. If these assumptions are violated, a stochastic approach to nonlinear modeling may be more appropriate, and it is natural to include noise terms in the above equations. For a stochastic approach to the nonlinear modeling of input­ output systems, see Billings [1]. Under the above deterministic assumptions, we are interested in the following theoretical questions. Firstly, how should chaos be defined and quantified for the system (I)? If the input time series u(t) is periodic, this question trivially reduces to that of autonomous systems by considering the appropriate time-T Poincare map. However, we are mostly interested in the case where u(t) is a random time series. If u(t) is random, then x(t) is random, so neither time series by itselfis chaotic. Hence this must be a question about the structure of the pair of time series u(t), x(t). We will address this question is Section 2. Secondly, if we wish to construct a nonlinear deterministic model of the form (3), how many lags m and I should be chosen for a system of dimension d? In the case of autonomous systems, this question has been addressed by Takens [14]; see also Sauer et al. [13]. We are also interested in how accurate such a model is likely to be as a function of the length of the time series available to construct it, and the dimension d of the underlying dynamical system. In the case of autonomous systems, this question has been addressed by Farmer and Sidorowich [7]; see also Casdagli [2]. We will investigate these questions for input­ output systems in Section 3. Finally, we summarize the conclusions in Section 4. 2 2 Chaos in Input-Output Systems In this section we give a definition of chaos for input-output systems. We also in­ vestigate the usefulness of this definition in quantifying predictability for a numerical example. For simplicity we will assume that time is discrete, so that Equation (1) is replaced by Sn+l = j(Sn' un) (4) Xn+l = h( Sn+l) (5) All of the results in this section generalize naturally to continuous time input-output systems by using results about Liapunov exponents in continuous time autonomous systems; see [6, 16J. 2.1 Definition of the largest Liapunov exponent Suppose the system (4) is initialized at two slightly different states, and in both cases is subjected to the same sequence of inputs Un' Then if the system states diverge exponentially in time, we say the input-output system is chaotic, and the rate of divergence is given by the largest Liapunov exponent. The largest Liapunov exponent quantifies the degree to which the system is predictable in the long term, assuming that the input time series is always observed. This observability assumption is satisfied in many of the applications mentioned in the introduction. For example, in vibration testing, one may only be able to observe the state of the system at rare intervals in the past due to measurement problems, and desire to predict the present state of the system given a sequence of inputs to the system. The assumption of having the input sequence available is also relevant for problems of reducing noise on output sequences observed in the past. Note that if the input time series is random and unobserved, the system is unpredictable even in the short term. We now make the above notions more precise. We will only be concerned with the largest Liapunov exponent, which is defined as follows. Let D j(s, u) denote the derivative of j at S with u held constant. Let D jT denote the matrix product T-1 DjT(so,uO,u,UT_1) = II Dj(Si,Ui) (6) i;:;O where the Si are generated from (4) starting from a given initial state so. Then the largest Liapunov exponent A, is defined by (7), where ds is an arbitrary initial vector, and II . II denotes the Euclidean norm. A, = lim .!:..log(11 DFds II / II ds II) (7) T~oo T If Al > 0 we say the system is chaotic. 3 The above definition of the Liapunov exponent Al at first sight depends on the initial state So, the sequence of inputs uo, Ul, .. and the tangent vector ds. However, suppose that the sequence of inputs is drawn from a realization of a stationary ran­ dom or deterministic process. Then in the case of random inputs, by multiplicative ergodic theorems (see [6]), the limit (7) exists with probability one, and depends only on the ergodic invariant measure to which the initial state So is attracted. This in­ variant measure is often unique, for example in the case of Gaussian inputs. In the case of deterministic inputs, although there are no general theorems, it is observed numerically that the limit (7) exists, and depends only on the basin of attraction in which So lies. The above definition coincides with the definition of Liapunov exponents for ran­ domly driven dynamical systems with unobserved inputs [6]. In this case, the input is assumed to be small, and models noise perturbations. However, if the inputs are observed, the largest Liapunov exponent may be used to describe the divergence of trajectories even for large amplitude inputs as follows. Let So and s~ denote two close initial conditions, subjected to the same sequence Uo, .. , UT_l of inputs. Then the divergence of trajectories will be described for moderate T by (8) Hence using (7) we obtain (9) We now illustrate the above ideas with the randomly driven Ikeda map where f is taken to be f(x,y,u) = 1 + a(xcost -ysint) +u,a(xsint +ycost) (10) 2 where t = 0.4 - 6.0/(1 + x + y2), a = 0.7 and the inputs Un are independently identically distributed (IID) Gaussians with variance 1]2. Figure 2 illustrates the de­ pendence of the Liapunov exponent Al on the noise level 1]. The Liapunov exponent was computed numerically using 10 5 iterates for each value of 1], with the QR al­ gorithm described in [6] to avoid overflow. Observe that a smooth transition from chaotic to non-chaotic behavior occurs at 1] "'" 0.95. Unlike autonomous systems, it is impossible to locate this transition by inspection of the invariant measure, which in randomly driven systems is always smooth.
Load more

Recommended publications
  • Do Chaos-Based Communication Systems Really Transmit Chaotic Signals?
    Signal Processing 108 (2015) 412–420 Contents lists available at ScienceDirect Signal Processing journal homepage: www.elsevier.com/locate/sigpro Do chaos-based communication systems really transmit chaotic signals? Renato Candido a, Diogo C. Soriano b, Magno T.M. Silva a,n, Marcio Eisencraft a a University of São Paulo, Brazil b Universidade Federal do ABC, Brazil article info abstract Article history: Many communication systems based on the synchronism of chaotic systems have been Received 2 May 2014 proposed as an alternative spread spectrum modulation that improves the level of privacy Received in revised form in data transmission. However, depending on the map and on the encoding function, the 24 September 2014 transmitted signal may cease to be chaotic. Therefore, the sensitive dependence on initial Accepted 2 October 2014 conditions, which is one of the most interesting properties for employing chaos in Available online 14 October 2014 telecommunications, may disappear. In this paper, we numerically analyze the chaotic Keywords: nature of signals modulated using a system that employs the Ikeda map. Additionally, we Chaos propose changes in the communication system in order to guarantee that the modulated Nonlinear systems signals are in fact chaotic. Chaos-based communication systems & 2014 Elsevier B.V. All rights reserved. Ikeda map Attractors 1. Introduction diverges from the original one. These three properties all together are necessary for a signal to be called chaotic Non-linear systems and chaos have been applied in all and are the basis for the alleged advantages of using chaos areas of engineering [1]. This fact is particularly true when in communications, as an improvement in security [15].
    [Show full text]
  • Efficient Method for Detection of Periodic Orbits in Chaotic Maps And
    Efficient Method for Detection of Periodic Orbits in Chaotic Maps and Flows A Thesis by Jonathan Crofts in partial fulfillment of the requirements for the degree of Doctor of Philosophy. Department of Mathematics University of Leicester March 2007. c Jonathan J Crofts, 2007. arXiv:0706.1940v2 [nlin.CD] 13 Jun 2007 Acknowledgements I would like to thank Ruslan Davidchack, my supervisor, for his many suggestions and constant support and understanding during this research. I am also thankful to Michael Tretyakov for his support and advice. Further, I would like to acknowledge my gratitude to the people from the Department of Mathematics at the University of Leicester for their help and support. Finally, I would like to thank my family and friends for their patience and sup- port throughout the past few years. In particular, I thank my wife Lisa and my daughter Ellen, without whom I would have completed this research far quicker, but somehow, it just would not have been the same. At this point I would also like to reassure Lisa that I will get a real job soon. Leicester, Leicestershire, UK Jonathan J. Crofts 31 March 2007 i Contents Abstract iv List of figures v List of tables vi 1 Introduction 1 1.1 History,theoryandapplications . 1 1.2 Periodicorbits............................... 3 1.2.1 Periodicorbittheory . 4 1.2.2 Efficient detection of UPOs . 6 1.3 Extendedsystems............................. 9 1.4 Anoteonnumerics............................ 10 1.4.1 Intervalarithmetic . 12 1.5 Overview.................................. 14 1.6 Thesisresults ............................... 16 2 Conventional techniques for detecting periodic orbits 18 2.1 Specialcases................................ 18 2.1.1 One-dimensionalmaps .
    [Show full text]
  • Transformations)
    TRANSFORMACJE (TRANSFORMATIONS) Transformacje (Transformations) is an interdisciplinary refereed, reviewed journal, published since 1992. The journal is devoted to i.a.: civilizational and cultural transformations, information (knowledge) societies, global problematique, sustainable development, political philosophy and values, future studies. The journal's quasi-paradigm is TRANSFORMATION - as a present stage and form of development of technology, society, culture, civilization, values, mindsets etc. Impacts and potentialities of change and transition need new methodological tools, new visions and innovation for theoretical and practical capacity-building. The journal aims to promote inter-, multi- and transdisci- plinary approach, future orientation and strategic and global thinking. Transformacje (Transformations) are internationally available – since 2012 we have a licence agrement with the global database: EBSCO Publishing (Ipswich, MA, USA) We are listed by INDEX COPERNICUS since 2013 I TRANSFORMACJE(TRANSFORMATIONS) 3-4 (78-79) 2013 ISSN 1230-0292 Reviewed journal Published twice a year (double issues) in Polish and English (separate papers) Editorial Staff: Prof. Lech W. ZACHER, Center of Impact Assessment Studies and Forecasting, Kozminski University, Warsaw, Poland ([email protected]) – Editor-in-Chief Prof. Dora MARINOVA, Sustainability Policy Institute, Curtin University, Perth, Australia ([email protected]) – Deputy Editor-in-Chief Prof. Tadeusz MICZKA, Institute of Cultural and Interdisciplinary Studies, University of Silesia, Katowice, Poland ([email protected]) – Deputy Editor-in-Chief Dr Małgorzata SKÓRZEWSKA-AMBERG, School of Law, Kozminski University, Warsaw, Poland ([email protected]) – Coordinator Dr Alina BETLEJ, Institute of Sociology, John Paul II Catholic University of Lublin, Poland Dr Mirosław GEISE, Institute of Political Sciences, Kazimierz Wielki University, Bydgoszcz, Poland (also statistical editor) Prof.
    [Show full text]
  • Chaotic Scattering in Solitary Wave Interactions: a Singular Iterated-Map Description Roy H
    CHAOS 18, 023113 ͑2008͒ Chaotic scattering in solitary wave interactions: A singular iterated-map description Roy H. Goodman Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, New Jersey 07102, USA ͑Received 16 October 2007; accepted 7 March 2008; published online 7 May 2008͒ We derive a family of singular iterated maps—closely related to Poincaré maps—that describe chaotic interactions between colliding solitary waves. The chaotic behavior of such solitary-wave collisions depends on the transfer of energy to a secondary mode of oscillation, often an internal mode of the pulse. This map allows us to go beyond previous analyses and to understand the interactions in the case when this mode is excited prior to the first collision. The map is derived using Melnikov integrals and matched asymptotic expansions and generalizes a “multipulse” Melnikov integral. It allows one to find not only multipulse heteroclinic orbits, but exotic periodic orbits. The maps exhibit singular behavior, including regions of infinite winding. These maps are shown to be singular versions of the conservative Ikeda map from laser physics and connections are made with problems from celestial mechanics and fluid mechanics. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2904823͔ Solitary waves are solutions to time-dependent partial generally lose their distinct identities and merge into a single differential equations (PDEs) which can be described as larger bump. At the opposite extreme are completely inte- profiles of unchanging spatial shape that move at con- grable systems whose solitary waves ͑solitons͒ survive col- stant velocity. A fundamental question is what happens lisions with their identities intact, due to a linear structure when two of them collide.
    [Show full text]
  • Intermittent Large Deviation of Chaotic Trajectory in Ikeda Map: Signature of Extreme Events Arnob Ray,1 Sarbendu Rakshit,1 Dibakar Ghosh,1, A) and Syamal K
    Intermittent large deviation of chaotic trajectory in Ikeda map: Signature of extreme events Arnob Ray,1 Sarbendu Rakshit,1 Dibakar Ghosh,1, a) and Syamal K. Dana2, 3 1)Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, India 2)Department of Mathematics, Jadavpur University, Kolkata 700032, India 3)Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland We notice signatures of extreme events-like behavior in a laser based Ikeda map. The trajectory of the system occasionally travels a large distance away from the bounded chaotic region, which appears as intermittent spiking events in the temporal dynamics. The large spiking events satisfy the conditions of extreme events as usually observed in dynamical systems. The probability density function of the large spiking events shows a long-tail distribution con- sistent with the characteristics of rare events. The inter-event intervals obey a Poisson-like distribution. We locate the parameter regions of extreme events in phase diagrams. Furthermore, we study two Ikeda maps to explore how and when extreme events terminate via mutual interaction. A pure diffusion of in- formation exchange is unable to terminate extreme events where synchronous occurrence of extreme events is only possible even for large interaction. On the other hand, a threshold-activated coupling can terminate extreme events above a critical value of mutual interaction. PACS numbers: 89.75.Fb, 05.45.a Rare and recurrent large amplitude deviations of normally bounded dynamics are seen in many systems. Such occasional large amplitude spiking events are larger than a nominal value and their statistical distribution of occurrence shows qualitative similarities, in dynamical sense, with data records of natural disas- ters, rogue waves, tsunami, flood and share market crashes.
    [Show full text]
  • Oct 0 3 2007 Barker
    Discriminating Noise from Chaos in Heart Rate Variability: Application to Prognosis in Heart Failure by Natalia M. Arzeno S.B. Electrical Science and Engineering Massachusetts Institute of Technology, 2006 Submitted to the Department of Electrical Engineering and Computer Science in Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Electrical Engineering and Computer Science at the Massachusetts Institute of Technology May 2007 Copyright 2007 Natalia M. ArzenoV. All rights reserved. The author hereby grants to M.I.T. permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole and in part in any medium now known or hereafter created. Author Department of Electrica Engineering and Computer Science May 25, 2007 Certified by Chi-Sang Poon V arch Scientist i Supervisor Accepted by thur C. Smith 1 Engineering Chairman, Department Committee on Graduate Theses MASSACHUSETTS INS E OF TECHNoLOGy OCT 0 3 2007 BARKER L IRA RIdS 2 Discriminating Noise from Chaos in Heart Rate Variability: Application to Prognosis in Heart Failure by Natalia M. Arzeno Submitted to the Department of Electrical Engineering and Computer Science May 25, 2007 In Partial Fulfillment of the Requirements for the Degree of Master of Engineering in Electrical Engineering and Computer Science ABSTRACT This thesis examines two challenging problems in chaos analysis: distinguishing deterministic chaos and stochastic (noise-induced) chaos, and applying chaotic heart rate variability (HRV) analysis to the prognosis of mortality in congestive heart failure (CHF). Distinguishing noise from chaos poses a major challenge in nonlinear dynamics theory since the addition of dynamic noise can make a non-chaotic nonlinear system exhibit stochastic chaos, a concept which is not well-defined and is the center of heated debate in chaos theory.
    [Show full text]
  • Pseudo-Orbit Data Assimilation. Part I: the Perfect Model Scenario
    Hailiang Du and Leonard A. Smith Pseudo-orbit data assimilation. Part I: the perfect model scenario Article (Published version) (Refereed) Original citation: Du, Hailiang and Smith, Leonard A. (2014) Pseudo-orbit data assimilation. part I: the perfect model scenario. Journal of the Atmospheric Sciences, 71 (2). pp. 469-482. ISSN 0022-4928 DOI: 10.1175/JAS-D-13-032.1 © 2014 American Meteorological Society This version available at: http://eprints.lse.ac.uk/55849/ Available in LSE Research Online: July 2014 LSE has developed LSE Research Online so that users may access research output of the School. Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Users may download and/or print one copy of any article(s) in LSE Research Online to facilitate their private study or for non-commercial research. You may not engage in further distribution of the material or use it for any profit-making activities or any commercial gain. You may freely distribute the URL (http://eprints.lse.ac.uk) of the LSE Research Online website. VOLUME 71 JOURNAL OF THE ATMOSPHERIC SCIENCES FEBRUARY 2014 Pseudo-Orbit Data Assimilation. Part I: The Perfect Model Scenario HAILIANG DU AND LEONARD A. SMITH Centre for the Analysis of Time Series, London School of Economics and Political Science, London, United Kingdom (Manuscript received 26 January 2013, in final form 21 October 2013) ABSTRACT State estimation lies at the heart of many meteorological tasks. Pseudo-orbit-based data assimilation provides an attractive alternative approach to data assimilation in nonlinear systems such as weather forecasting models.
    [Show full text]
  • Applied .Chaos: Quantifying Complex Systems
    APPLIED .C HAOS: QUANTIFYING COMPLEX SYSTEMS LEONARD A. SMITI!(l] Department of Applied Mathematics and Theoretical Physics University of Cambridge Cambridge CB3 9EW, U.K. Abstract This contribution discusses dynamic reconstructions <1Ild their application to the identification of, quantification of and discrimination between complex systems. Phase space reconstructions which reproduce the flow of dynamical systems in time are con­ structed from (multiple probe) time series data. These dynamic reconstructions are used to quantify the unknown system. In the case of chaotic systems, this is ac­ complished through the isolation of unstable periodic orbits. The method is applied to data from the Ikeda map, where it is shown that the data requirements of this approach are modest relative to those required for other types of analysis. The ap­ plication of this approach to systems where the underlying dynamics are stochastic is also discussed . 1. Introduction In recent years there has been a rapid increase in the application of nonlinear dynamical systems ideas to the analysis of complex time series. These ideas have provided a new paradigm which has proven useful in many fields of study. Originally, reconstructions were "static" in that they attempted to reconstruct the geometry from a time seri es [1] . Unfortunately, methods of analysis developed for the study of low dimensional low noise systems h ave sometimes been applied to data sets with whi ch they cannot produce meaningful results. This is particularly true of dimension calcu­ lations [2] where many results may be disqualified on sim ple geometrin t! constraints [3]. While much work has been done on determining the reliabi lity of such algorithms [4], a rrl<ljOf draw back of these approaches is t.haL they fail in suut.lc ways.
    [Show full text]
  • EXPLOSION POINTS in SKEW MAPS by Anne Stefanie Costolanski A
    EXPLOSION POINTS IN SKEW MAPS by Anne Stefanie Costolanski A Thesis Submitted to the Graduate Faculty of George Mason University in Partial Fulfillment of The Requirements for the Degree of Master of Science Mathematics Committee: Dr. Evelyn Sander, Thesis Director Dr. Kathleen Alligood, Committee Member Dr. Timothy Sauer, Committee Member Dr. Klaus Fischer, Department Chairperson Dr. Peter Becker, Associate Dean for Graduate Studies, College of Science Dr. Vikas Chandhoke, Dean, College of Science Date: Summer Semester 2007 George Mason University Fairfax, VA Explosion Points in Skew Maps A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at George Mason University By Anne Stefanie Costolanski Bachelor of Science Virginia Polytechnic and State University, 1988 Director: Dr. Evelyn Sander, Professor Department of Mathematics Summer Semester 2007 George Mason University Fairfax, VA ii Acknowledgments I would like to thank my thesis advisor, Dr. Evelyn Sander, for her assistance, patience, and faith in my ability to do this work. Her encouragement throughout the past two years has been crucial in helping me complete this thesis and continue forward with my mathematics education. I would also like to acknowledge the many friends that I have made over my lifetime, but in particular those who have helped me through the past seven years. You are more than just friends, you are my family, and I am eternally grateful for your unwavering support and unconditional love. iii Table of Contents Page Abstract....................................... vi 1 Introduction .................................. 1 1.1 DynamicalSystems............................ 1 1.2 StabilityAnalysis ............................. 2 1.3 ChaoticBehavior ............................. 5 1.4 OneParameterFamilies ........................
    [Show full text]
  • An Existence of Simple Choreographies for N-Body Problem - a Computer Assisted Proof
    AN EXISTENCE OF SIMPLE CHOREOGRAPHIES FOR N-BODY PROBLEM - A COMPUTER ASSISTED PROOF TOMASZ KAPELA AND PIOTR ZGLICZYNSKI´ Abstract. We consider a question of finding a periodic solution for the planar Newtonian N-body problem with equal masses, where each body is travelling along the same closed path. We provide a computer assisted proof for the following facts: local uniqueness and convexity of Chenciner and Montgomery Eight, an existence (and local uniqueness) for Gerver’s SuperEight for 4-bodies and a doubly symmetric linear chain for 6-bodies. 1. Introduction In this paper we consider the problem of finding periodic solution to the N-body problem in which all N masses travel along a fixed curve in the plane. The N-body problem with N equal unit masses is given by a differential equation X q ¡ q (1.1)q ¨ = j i i r3 j6=i ij n where qi 2 R , i = 1; :::; N, rij = jjqi ¡ qjjj. The gravitational constant is taken equal to 1. We consider planar case (n=2) only, we set qi = (xi; yi),q ˙i = (vi; ui). Using this we can express (1.1) by: 8 P xj ¡xi > v˙i = j6=i r3 > ij < P yj ¡yi u˙i = j6=i 3 (1.2) rij > x˙ = v :> i i y˙i = ui Recently, this problem received a lot attention in literature see [M, CM, CGMS, S1, S2, S3, MR] and papers cited there. By a simple choreography [S1, S2] we mean a collision-free solution of the N-body problem in which all masses move on the same curve with a constant phase shift.
    [Show full text]
  • A Period-Doubling Cascade Precedes Chaos for Planar Maps Evelyn Sander and James A
    A period-doubling cascade precedes chaos for planar maps Evelyn Sander and James A. Yorke Citation: Chaos 23, 033113 (2013); doi: 10.1063/1.4813600 View online: http://dx.doi.org/10.1063/1.4813600 View Table of Contents: http://chaos.aip.org/resource/1/CHAOEH/v23/i3 Published by the AIP Publishing LLC. Additional information on Chaos Journal Homepage: http://chaos.aip.org/ Journal Information: http://chaos.aip.org/about/about_the_journal Top downloads: http://chaos.aip.org/features/most_downloaded Information for Authors: http://chaos.aip.org/authors CHAOS 23, 033113 (2013) A period-doubling cascade precedes chaos for planar maps Evelyn Sander1,a) and James A. Yorke2,b) 1Department of Mathematical Sciences, George Mason University, 4400 University Dr., Fairfax, Virginia 22030, USA 2University of Maryland, College Park, Maryland 20742, USA (Received 25 April 2013; accepted 27 June 2013; published online 22 July 2013) A period-doubling cascade is often seen in numerical studies of those smooth (one-parameter families of) maps for which as the parameter is varied, the map transitions from one without chaos to one with chaos. Our emphasis in this paper is on establishing the existence of such a cascade for many maps with phase space dimension 2. We use continuation methods to show the following: under certain general assumptions, if at one parameter there are only finitely many periodic orbits, and at another parameter value there is chaos, then between those two parameter values there must be a cascade. We investigate only families that are generic in the sense that all periodic orbit bifurcations are generic.
    [Show full text]
  • Rigorous Investigations of Ikeda Map by Means of Interval Arithmetic
    Rigorous investigations of Ikeda map by means of interval arithmetic Zbigniew Galias Department of Electrical Engineering, University of Mining and Metallurgy al. Mickiewicza 30, 30{059 Krak´ow, Poland e-mail: [email protected] Abstract. In this work we present various tools for studying dynamical systems implemented in interval arithmetic. The methods include an algorithm for finding all low-period cycles enclosed in a specified region, finding an upper bound of the invariant and nonwandering part of a given set, finding a lower bound of the basin of attraction of a stable periodic orbit, and proving the existence of symbolic dynamics of a given type embedded in the map. Using these techniques a detailed study of the behavior of the Ikeda map for different parameter values is performed. AMS classification scheme numbers: 37E30, 37B10, 37B40, 37C25 Submitted to: Nonlinearity Rigorous investigations of Ikeda map 2 1. Introduction Basic tools for the investigations of nonlinear systems are laboratory experiments and simulations. In simulations periodic orbits are usually found using some version of the Newton method, complex trajectories are observed by simply iterating the map on a computer in machine precision. Many of these results are left without a rigorous proof. There is no guarantee that there exist a true trajectory, which stays near the one generated by a computer. For chaotic systems, which exhibit sensitive dependence on initial conditions this problem is especially important. When we iterate the map using a computer the inevitable rounding errors cause that the trajectory generated by a computer and the true one diverge exponentially and after a certain number of iterations become uncorrelated.
    [Show full text]