DOCSLIB.ORG

Mathematical Problems of Nonlinear Dynamics: a Tutorial

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Mathematical Problems of Nonlinear Dynamics: a Tutorial Mathematical problems of nonlinear dynamics: A tutorial Leonid Shilnikov Abstract We review the theory of nonlinear systems, especially that of strange attractors, and give its perspectives. Of a special attention are the recent results concerning hyperbolic attractors and features of high-dimensional systems in the Newhouse regions. We present an example of a “wild” strange attractor of the topological dimension three. 1 Content 1. Introduction 2. Basic notions of the theory of dynamical systems 3. The Andronov-Pontryagin theorem. Morse-Smale systems 4. Poincar´ehomoclinic structures 5. Structurally stable systems 6. The bifurcation theory. Appearance of hyperbolic attractors 7. Structurally unstable systems. Wild hyperbolic sets. Newhouse regions 8. Lorenz attractor 9. Quasiattractors. Non-transverse homoclinic curves 10. Example of wild strange attractor 2 1 Introduction The early 60th is the beginning of the intensive development of the theory of high-dimensional dynamical systems. Within a short period of time Smale [66, 67] had established the basics of the theory of structurally unstable systems with the complex behaviour of trajectories, the theory which we now know as the hyperbolic theory. In essence, a new mathematical discipline with its own terminology, notions etc, has been created, which at the same time interacts actively with other mathematical disciplines. Here we must emphasize the role of the qualitative theory of differential equation (QTODEs). In fact, this theory provides a foundation for investigating many problems of natural sciences and engineering which have a nonlinear dynamics origin. On the other hand the qualitative theory of differential equations itself takes new ideas from nonlinear dynamics. The usefulness and the necessity of such a synthesis were clear for such scientists with the broad vision on science as Poincar´eand Andronov. All this makes the QTODEs especially attractive and practical. Its achievements have led to one of the brightest scientific discoveries of the XX century — dy- namical chaos. Since the moment of the discovery of dynamical chaos along with such customary dynamical regimes as stationary states, self-oscillations and modulations, chaotic oscillations entered the science. If the mathemati- cal images of the formers are equilibrium states, periodic orbits and tori with quasiperiodic trajectories, then the adequate image of dynamical chaos is a strange attractor, i.e., an attractive limiting set with the unstable behaviour of its trajectories. Those attractors that persist this property under small smooth perturbations will interest us in this review. Namely, such attractors have been predicted by the hyperbolic theory of high-dimensional dynamical systems. However, the role and the significance of strange attractors were not accepted by researchers of certain scientific directions, in particular of turbulence, for a sufficiently long time. There were a few reasons for that. The hyperbolic theory had examples of strange attractors but their structure were so topologically 3 complex that did not allow one to imagine rather simple scenarios of their origination which is very important for nonlinear dynamics, which deals with models described by differential equations. On the other hand, those “strange attractors” observed in concrete models were not hyperbolic attractors in the strict meaning of these words. Most of them, possessing all the properties of a “qenuine” strange attractor, had stable periodic orbits. This gave a chance to argue that the observable chaotic behaviour is intermittent. Here, we have to bear in mind that when speaking about dynamical systems we are interested not in the character of a solution over some bounded period of time but in the information on its limiting behaviour when time increases to infinity. Note also strange attractors that have hyperbolic subsets co-existing with stable long periodic orbits of very narrow and tortious attraction basin, the so-called quasi- 1attractors [10]. The breakthrough came in the mid 70-ths with the appearance of a “simple” low-dimensional model x˙ = σ(x y), y˙ = rx− y− xz, z˙ = bz−+ xy− − in which Lorenz had discovered numerically in 1962 a chaotic behaviour in the trajectories. A detailed analysis carried out by mathematicians revealed the existence of a strange attractor, not hyperbolic but non-structurally stable. Nevertheless, the main feature — the instability of the behaviour of trajectories under small smooth perturbations of the system — of this attractor persists. Such attractors, which contain a single equilibrium state of the saddle type, will be henceforth be called Lorenz(ian) attractors. The second remarkable fact related to these attractors is that the Lorenz attractor may be generated on the route of a finite number of rather simple observable bifurcations from systems with trivial dynamics. Since that time the phenomenon of dynamical chaos was “almost legislated”. In this breakthrough, the fact that the Lorenz model came from hydrodynamics played not the least but a primary role. The topological dimension of the Lorenz attractor, regardless of the dimen- 1stochastic 4 sion of the associated concrete system, is always two and its fractal dimension is less than three. At the same time, researchers who deal with extended sys- tems often observe chaotic regimes of presumably much higher dimensions. It is customary then to say that hyperchaos occurs. But which attractors describe hyperchaos? Are they strange attractors or quasiattractors? In principal, the hyperbolic theory predicts the possibility of the existence of strange attractor of any finite dimensions. The tragedy is that nobody has observed known hy- perbolic attractors in nonlinear dynamics since the moment of their invention. There has been some progress recently: the author and Turaev [79] have proved that a number of hyperbolic attractors (structurally like the Smale-Williams solenoids and the Anosov tori) may be obtained through one global bifurcation of the disappearance of a stable periodic orbit or of an invariant torus with a quasi-periodic trajectory on it. They have also discovered a principally new type of strange attractor — the so-called wild strange attractors. Their distinc- tion from the known attractors is that they contain an equilibrium state of the saddle-focus type as well as saddle periodic trajectories of various types, namely, dimensions of invariant manifolds of the co-existing trajectories may be equal both two and three. Moreover, the region of the existence of such an attractor is a region of everywhere dense structural instability due to homoclinic tangencies. Thus, the above phenomenon poses principally new problems for ergodic theory. Due to the co-existence of the trajectories of various types the wild attractors, as well as Lorenz-like attractors, are the pseudo-hyperbolic attractors. Since the notion of pseidohyperbolicity, which plays a dominating role in the theory of structurally unstable strange attractors, will be used below. So let us stop here and discuss it in detail. Consider a smooth n-dimensional dynamical system x˙ = X(x), in a bounded region D which satisfies the following conditions: 1. On its boundary ∂D the vector flow goes inward D. This implies that for any point x ∂D either an entire trajectory or a semitrajectory is defined ∈ that passes through the point x. 5 2. A pseudo-hyperbolicity takes place in D. This implies that at each point x D the tangent space, invariant with respect to the associated lin- ∈ earized flow, may be decomposed as a direct sum of subspaces N1 and N2, depending continuously on the point x so that the maximal Lyapunov ex- ponent, corresponding to N1, is strongly less than any Lyapunov exponent corresponding to N2. In other words, the associated variational equation can be represented in the form ξ˙ = A1(t)ξ, η˙ = A2(t)η, where the contraction in ξ is stronger than the contraction in η. 3. The linearized semiflow is volume-expanding V const eσtV , σ> 0. t ≥ 0 Note that the property of the pseudo-hyperbolicity persists under small smooth perturbations, as does the property of the exponential expansion of volumes in N2. Due to the above requirements there will exist at least one strange attractor in the region D. Note that this suggested criterion on the existence of a pseudo- hyperbolic attractor in D is formulated relatively simply, but, similarly the principle of the contraction mappings, its real verification in concrete systems will not be trivial. Let us return to the problem of quasi-attractors. In both situation, in the case of quasi-attractors and in the case of wild strange attractors, the reason of complexity is the presence of structurally unstable Poincar´ehomoclinic curves, i.e., bi-asymptotic trajectories to a saddle periodic orbit, along which its stable and unstable manifolds have a non-transverse contact. This type of homo- clinic trajectories is also responsible for the existence in the space of dynamical systems of regions of everywhere dense structural instability — the so-called Newhouse regions in which systems with homoclinic tangencies are dense. Un- der certain conditions systems with infinitely many stable periodic orbits are also dense in the Newhouse regions. At least this is always true for three- dimensional systems with negative divergence. The peculiarity of such a set of 6 stable periodic orbits is that it cannot be separated in a quasiattractor from the co-existing hyperbolic subset to which these periodic orbits accumulate. In three-dimensional
Load more

Recommended publications
  • Chapter 2 Structural Stability
    Chapter 2 Structural stability 2.1 Denitions and one-dimensional examples A very important notion, both from a theoretical point of view and for applications, is that of stability: the qualitative behavior should not change under small perturbations. Denition 2.1.1: A Cr map f is Cm structurally stable (with 1 m r ∞) if there exists a neighbour- hood U of f in the Cm topology such that every g ∈ U is topologically conjugated to f. Remark 2.1.1. The reason that for structural stability we just ask the existence of a topological conju- gacy with close maps is because we are interested only in the qualitative properties of the dynamics. For 1 1 R instance, the maps f(x)= 2xand g(x)= 3xhave the same qualitative dynamics over (and indeed are topologically conjugated; see below) but they cannot be C1-conjugated. Indeed, assume there is a C1- dieomorphism h: R → R such that h g = f h. Then we must have h(0) = 0 (because the origin is the unique xed point of both f and g) and 1 1 h0(0) = h0 g(0) g0(0)=(hg)0(0)=(fh)0(0) = f 0 h(0) h0(0) = h0(0); 3 2 but this implies h0(0) = 0, which is impossible. Let us begin with examples of non-structurally stable maps. 2 Example 2.1.1. For ε ∈ R let Fε: R → R given by Fε(x)=xx +ε. We have kFε F0kr = |ε| for r all r 0, and hence Fε → F0 in the C topology.
    [Show full text]
  • Fine Structure of Hyperbolic Diffeomorphisms, by A. A. Pinto, D
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 48, Number 1, January 2011, Pages 131–136 S 0273-0979(2010)01284-2 Article electronically published on May 24, 2010 Fine structure of hyperbolic diffeomorphisms,byA.A.Pinto,D.Rand,andF.Fer- reira, Springer Monographs in Mathematics, Springer-Verlag, Berlin, Heidelberg, 2009, xvi+354 pp., ISBN 978-3-540-87524-6, hardcover, US$129.00 The main theme of the book Fine Structures of Hyperbolic Diffeomorphisms,by Pinto, Rand and Ferreira, is the rigidity and flexibility of two-dimensional diffeo- morphisms on hyperbolic basic sets and properties of invariant measures that are related to the geometry of these invariant sets. In his remarkable article [23], Smale sets the foundations of the modern theory of dynamical systems. He defines the fundamental notion of hyperbolicity and relates it to structural stability. Let f be a smooth (at least C1) diffeomorphism of a compact manifold M. A hyperbolic set for f is a closed f-invariant subset Λ ⊂ M such that the tangent bundle of the manifold over Λ splits as a direct sum of two subbundles that are invariant under the derivative, and the derivative of the iterates of the map expands exponentially one of the bundles (the unstable bundle) and contracts exponentially the stable subbundle. These bundles are in general only continuous, but they are integrable. Through each point x ∈ Λ, there exists a one-to-one immersed submanifold W s(x), the stable manifold of x.This submanifold is tangent to the stable bundle at each point of intersection with Λ and is characterized by the fact that the orbit of each point y ∈ W s(x)isasymptotic to the orbit of x, and, in fact, the distance between f n(y)tof n(x)converges to zero exponentially fast.
    [Show full text]
  • The Theory of Filtrations of Subalgebras, Standardness and Independence
    The theory of filtrations of subalgebras, standardness and independence Anatoly M. Vershik∗ 24.01.2017 In memory of my friend Kolya K. (1933{2014) \Independence is the best quality, the best word in all languages." J. Brodsky. From a personal letter. Abstract The survey is devoted to the combinatorial and metric theory of fil- trations, i. e., decreasing sequences of σ-algebras in measure spaces or decreasing sequences of subalgebras of certain algebras. One of the key notions, that of standardness, plays the role of a generalization of the no- tion of the independence of a sequence of random variables. We discuss the possibility of obtaining a classification of filtrations, their invariants, and various links to problems in algebra, dynamics, and combinatorics. Bibliography: 101 titles. Contents 1 Introduction 3 1.1 A simple example and a difficult question . .3 1.2 Three languages of measure theory . .5 1.3 Where do filtrations appear? . .6 1.4 Finite and infinite in classification problems; standardness as a generalization of independence . .9 arXiv:1705.06619v1 [math.DS] 18 May 2017 1.5 A summary of the paper . 12 2 The definition of filtrations in measure spaces 13 2.1 Measure theory: a Lebesgue space, basic facts . 13 2.2 Measurable partitions, filtrations . 14 2.3 Classification of measurable partitions . 16 ∗St. Petersburg Department of Steklov Mathematical Institute; St. Petersburg State Uni- versity Institute for Information Transmission Problems. E-mail: [email protected]. Par- tially supported by the Russian Science Foundation (grant No. 14-11-00581). 1 2.4 Classification of finite filtrations . 17 2.5 Filtrations we consider and how one can define them .
    [Show full text]
  • The Entropy Function for Non Polynomial Problems and Its Applications for Turing Machines
    Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 30 January 2020 doi:10.20944/preprints202001.0360.v1 The entropy function for non polynomial problems and its applications for turing machines. Matheus Santana Harvard Extension School [email protected] Abstract We present a general process for the halting problem, valid regardless of the time and space computational complexity of the decision problem. It can be interpreted as the maximization of entropy for the utility function of a given Shannon-Kolmogorov-Bernoulli process. Applications to non-polynomials prob- lems are given. The new interpretation of information rate proposed in this work is a method that models the solution space boundaries of any decision problem (and non polynomial problems in general) as a communication channel by means of Information Theory. We described a sort method that order objects using the intrinsic information content distribution for the elements of a constrained solution space - modeled as messages transmitted through any communication systems. The limits of the search space are defined by the Kolmogorov-Chaitin complexity of the sequences encoded as Shannon-Bernoulli strings. We conclude with a discussion about the implications for general decision problems in Turing machines. Keywords: Computational Complexity, Information Theory, Machine Learning, Computational Statistics, Kolmogorov-Chaitin Complexity; Kelly criterion 1. Introduction Consider a Shanon-Bernoulli process P defined as a sequence of independent binary random variables X1, X2, X3 . Xn. For each element the value can be Preprint submitted to Journal of Information and Computation January 29, 2020 © 2020 by the author(s). Distributed under a Creative Commons CC BY license.
    [Show full text]
  • Writing the History of Dynamical Systems and Chaos
    Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as
    [Show full text]
  • A Structural Approach to Solving the 6Th Hilbert Problem Udc 519.21
    Teor Imovr.taMatem.Statist. Theor. Probability and Math. Statist. Vip. 71, 2004 No. 71, 2005, Pages 165–179 S 0094-9000(05)00656-3 Article electronically published on December 30, 2005 A STRUCTURAL APPROACH TO SOLVING THE 6TH HILBERT PROBLEM UDC 519.21 YU. I. PETUNIN AND D. A. KLYUSHIN Abstract. The paper deals with an approach to solving the 6th Hilbert problem based on interpreting the field of random events as a partially ordered set endowed with a natural order of random events obtained by formalization and modification of the frequency definition of probability. It is shown that the field of events forms an atomic generated, complete, and completely distributive Boolean algebra. The probability distribution of the field of events generated by random variables is studied. It is proved that the probability distribution generated by random variables is not a measure but only a finitely additive function of events in the case of continuous random variables (both rational- and real-valued). Introduction In 1900 in his lecture [1] David Hilbert formulated the problem of axiomatization of probability theory. Here is the corresponding quote from his lecture: “The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. As to the axioms of the theory of probabilities, it seems to me desirable that their logical investigation should be accompanied by a rigorous and satisfactory development of the method of mean values in mathematical physics, and in particular in the kinetic theory of gases.” As is well known [2], there is no generally accepted solution of this problem so far.
    [Show full text]
  • Nonlocally Maximal Hyperbolic Sets for Flows
    Brigham Young University BYU ScholarsArchive Theses and Dissertations 2015-06-01 Nonlocally Maximal Hyperbolic Sets for Flows Taylor Michael Petty Brigham Young University - Provo Follow this and additional works at: https://scholarsarchive.byu.edu/etd Part of the Mathematics Commons BYU ScholarsArchive Citation Petty, Taylor Michael, "Nonlocally Maximal Hyperbolic Sets for Flows" (2015). Theses and Dissertations. 5558. https://scholarsarchive.byu.edu/etd/5558 This Thesis is brought to you for free and open access by BYU ScholarsArchive. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of BYU ScholarsArchive. For more information, please contact [email protected], [email protected]. Nonlocally Maximal Hyperbolic Sets for Flows Taylor Michael Petty A thesis submitted to the faculty of Brigham Young University in partial fulfillment of the requirements for the degree of Master of Science Todd Fisher, Chair Lennard F. Bakker Christopher P. Grant Department of Mathematics Brigham Young University June 2015 Copyright c 2015 Taylor Michael Petty All Rights Reserved abstract Nonlocally Maximal Hyperbolic Sets for Flows Taylor Michael Petty Department of Mathematics, BYU Master of Science In 2004, Fisher constructed a map on a 2-disc that admitted a hyperbolic set not contained in any locally maximal hyperbolic set. Furthermore, it was shown that this was an open property, and that it was embeddable into any smooth manifold of dimension greater than one. In the present work we show that analogous results hold for flows. Specifically, on any smooth manifold with dimension greater than or equal to three there exists an open set of flows such that each flow in the open set contains a hyperbolic set that is not contained in a locally maximal one.
    [Show full text]
  • The Ergodic Theorem
    The Ergodic Theorem 1 Introduction Ergodic theory is a branch of mathematics which uses measure theory to study the long term be- haviour of dynamic systems. The central object of consideration is known as a measure-preserving system, a type of dynamic system where the evolution of the system preserves a measure. Definition 1: Let (X; M; µ) be a finite measure space and let T map X to itself. We say that T is a measure-preserving transformation if for every A 2 M, we have µ(A) = µ(T −1(A)). The quadruple (X; M; µ, T ) is called a measure-preserving system (m.p.s.). Measure-preserving systems arise in a variety of contexts, such as probability theory, informa- tion theory, and of course in the study of dynamical systems. However, ergodic theory originated from statistical mechanics. In this setting, T represents the evolution of the system through time. Given a measurable function f : X ! R, the series of values f(x); f(T x); f(T 2x)::: are the values of a physical observable at certain time intervals. Of importance in statistical mechanics is the long-term average of these observables: N−1 1 X f (x) = f(T kx) N N k=0 The Ergodic Theorem (also known as the Pointwise or Birkhoff Ergodic Theorem) is central to the study of averages such as fN in the limit as N ! 1. In this paper we aim to prove the theorem, and then discuss a few of its applications. Before we can state the theorem, we need another definition.
    [Show full text]
  • Centralizers of Anosov Diffeomorphisms on Tori
    ANNALES SCIENTIFIQUES DE L’É.N.S. J. PALIS J.-C. YOCCOZ Centralizers of Anosov diffeomorphisms on tori Annales scientifiques de l’É.N.S. 4e série, tome 22, no 1 (1989), p. 99-108 <http://www.numdam.org/item?id=ASENS_1989_4_22_1_99_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1989, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systé- matique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. scient. EC. Norm. Sup., 46 serie, t. 22, 1989, p. 99 a 108. CENTRALIZERS OF ANOSOV DIFFEOMORPHISMS ON TORI BY J. PALIS AND J. C. YOCCOZ ABSTRACT. — We prove here that the elements of an open and dense subset of Anosov diffeomorphisms on tori have trivial centralizers: they only commute with their own powers. 1. Introduction Let M be a smooth connected compact manifold, and Diff(M) the group of C°° diffeomorphisms of M endowed with the C°° topology. The diffeomorphisms which satisfy Axiom A and the (strong) transversality condition—every stable manifold intersects transversely every unstable manifold—form an open subset 91 (M) of Diff(M) and are, by Robbin [4] and a recent result of Mane [2], exactly the C1-structurally stable diffeomorphisms. We continue here the study, initiated in [3], of centralizers of diffeomorphisms in ^(M); the concepts we just mentioned are detailed there.
    [Show full text]
  • Structural Stability of Piecewise Smooth Systems
    STRUCTURAL STABILITY OF PIECEWISE SMOOTH SYSTEMS MIREILLE E. BROUCKE, CHARLES C. PUGH, AND SLOBODAN N. SIMIC´ † Abstract A.F. Filippov has developed a theory of dynamical systems that are governed by piecewise smooth vector fields [2]. It is mainly a local theory. In this article we concentrate on some of its global and generic aspects. We establish a generic structural stability theorem for Filip- pov systems on surfaces, which is a natural generalization of Mauricio Peixoto’s classic result [12]. We show that the generic Filippov system can be obtained from a smooth system by a process called pinching. Lastly, we give examples. Our work has precursors in an announcement by V.S. Kozlova [6] about structural stability for the case of planar Fil- ippov systems, and also the papers of Jorge Sotomayor and Jaume Llibre [8] and Marco Antonio Teixiera [17], [18]. Partially supported by NASA grant NAG-2-1039 and EPRI grant EPRI-35352- 6089.† STRUCTURAL STABILITY OF PIECEWISE SMOOTH SYSTEMS 1 1. Introduction Imagine two independently defined smooth vector fields on the 2- sphere, say X+ and X−. While a point p is in the Northern hemisphere let it move under the influence of X+, and while it is in the Southern hemisphere, let it move under the influence of X−. At the equator, make some intelligent decision about the motion of p. See Figure 1. This will give an orbit portrait on the sphere. What can it look like? Figure 1. A piecewise smooth vector field on the 2-sphere. How do perturbations affect it? How does it differ from the standard vector field case in which X+ = X−? These topics will be put in proper context and addressed in Sections 2-7.
    [Show full text]
  • Stability and Explanatory Significance of Some Simple Evolutionary Models
    Stability and Explanatory Significance of Some Simple Evolutionary Models Brian Skyrms University of California, Irvine 1. Introduction. The explanatory value of equilibrium depends on the underlying dynamics. First there are questions of dynamical stability of the equilibrium that are internal to the dynamical system in question. Is the equilibrium locally stable, so that states near to it stay near to it, or better, asymptotically stable, so that states near to it are carried to it by the dynamics? If not, we should not expect to see this equilibrium. But even if an equilibrium is asymptotically stable, that is no guarantee that the system will reach that equilibrium unless we know that the system's initial state is sufficiently close to the equilibrium. Global stability of an equilibrium, when we have it, gives the equilibrium a much more powerful explanatory role. An equilibrium is globally asymptotically stable if the dynamics carries every possible initial state in the interior of the state space to that equilibrium. If an equilibrium is globally stable, it can have explanatory value even when we are completely uncertain about the initial state of the system. Once questions of dynamical stability are answered with respect to the dynamical system in question, there is the further question of structural stability of that system itself. That is to say, are dynamical systems close to the one in question (in a sense to be made 1 precise) topologically equivalent to that system? If not, a slight mispecification of the model may make predictions that are drastically wrong. Structural stability is defined in terms of small changes in the model.
    [Show full text]
  • Stability and Bifurcation of Dynamical Systems
    STABILITY AND BIFURCATION OF DYNAMICAL SYSTEMS Scope: • To remind basic notions of Dynamical Systems and Stability Theory; • To introduce fundaments of Bifurcation Theory, and establish a link with Stability Theory; • To give an outline of the Center Manifold Method and Normal Form theory. 1 Outline: 1. General definitions 2. Fundaments of Stability Theory 3. Fundaments of Bifurcation Theory 4. Multiple bifurcations from a known path 5. The Center Manifold Method (CMM) 6. The Normal Form Theory (NFT) 2 1. GENERAL DEFINITIONS We give general definitions for a N-dimensional autonomous systems. ••• Equations of motion: x(t )= Fx ( ( t )), x ∈ »N where x are state-variables , { x} the state-space , and F the vector field . ••• Orbits: Let xS (t ) be the solution to equations which satisfies prescribed initial conditions: x S(t )= F ( x S ( t )) 0 xS (0) = x The set of all the values assumed by xS (t ) for t > 0is called an orbit of the dynamical system. Geometrically, an orbit is a curve in the phase-space, originating from x0. The set of all orbits is the phase-portrait or phase-flow . 3 ••• Classifications of orbits: Orbits are classified according to their time-behavior. Equilibrium (or fixed -) point : it is an orbit xS (t ) =: xE independent of time (represented by a point in the phase-space); Periodic orbit : it is an orbit xS (t ) =:xP (t ) such that xP(t+ T ) = x P () t , with T the period (it is a closed curve, called cycle ); Quasi-periodic orbit : it is an orbit xS (t ) =:xQ (t ) such that, given an arbitrary small ε > 0 , there exists a time τ for which xQ(t+τ ) − x Q () t ≤ ε holds for any t; (it is a curve that densely fills a ‘tubular’ space); Non-periodic orbit : orbit xS (t ) with no special properties.
    [Show full text]