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Shadowing in Dynamical Systems

Theory and Applications

verfasst von: Ken Palmer

Verlag: Springer US

Buchreihe : Mathematics and Its Applications

Enthalten in: Professional Book Archive

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Über dieses Buch

In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic.
It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic.
Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.

Inhaltsverzeichnis

Frontmatter

1. Hyperbolic Fixed Points of Diffeomorphisms and Their Stable and Unstable Manifolds

Abstract

The most common kind of dynamical system is that coming from a differential equation since it is most commonly differential equations which are used to model physical processes. However, as we shall see later in this book, the study of the dynamical properties of a differential equation can often be reduced to the study of the properties of a diffeomorphism. Diffeomorphisms are also of interest in themselves since they can be used as discrete models of physical and biological processes. Also in some ways they are easier to study than differential equations. For these reasons, we begin this book with the study of diffeomorphisms.

Ken Palmer

2. Hyperbolic Sets of Diffeomorphisms

Abstract

In the previous chapter, we discussed hyperbolic fixed points. In this chapter we want to consider more general invariant sets consisting of more than one point and perhaps infinitely many points. We give the appropriate definition of hyperbolicity for such sets and show that the continuity of the splitting into stable and unstable bundles follows from the other items in the definition. Then we expound the theory of exponential dichotomies for difference equations and use it to show that hyperbolic sets are expansive and that they are robust under perturbation.

Ken Palmer

3. Transversal Homoclinic Points of Diffeomorphisms and Hyperbolic Sets

Abstract

Up till now our only examples of hyperbolic sets have been hyperbolic fixed points and periodic orbits. In this chapter we give an example of a hyperbolic set which is infinite. Moreover, even though the set consists of only two distinct orbits, we shall show in Chapter 5 that in its neighbourhood the diffeomorphism has chaotic dynamics.

Ken Palmer

4. The Shadowing Theorem for Hyperbolic Sets of Diffeomorphisms

Abstract

In this chapter we prove the shadowing theorem for diffeomorphisms. Also we give two applications of it to hyperbolic sets. A further application follows in the next chapter.

Ken Palmer

5. Symbolic Dynamics Near a Transversal Homoclinic Point of a Diffeomorphism

Abstract

In Chapter 3 we showed that the set consisting of a hyperbolic fixed point of a diffeomorphism and an associated transversal homoclinic orbit is hyperbolic. In this chapter we use symbolic dynamics to describe all the orbits in a neighbourhood of this set. In particular, we shall show that the dynamics in such a neighbourhood is chaotic. Our main tool in establishing the symbolic dynamics is the shadowing theorem (Theorem 4.3).

Ken Palmer

6. Hyperbolic Periodic Orbits of Ordinary Differential Equations, Stable and Unstable Manifolds and Asymptotic Phase

Abstract

We wish to develop a theory for autonomous systems of ordinary differential equations analogous to the theory we have developed for diflfeomorphisms in Chapters 1 through 5. It turns out that the object analogous to the fixed point of a diflfeomorphism is a periodic solution rather than an equilibrium point. To some extent, we can reduce the study of a periodic solution to that of the fixed point of a diflfeomorphism by using the Poincaré map. However, first we begin by recalling a few elementary facts from the theory of ordinary differential equations.

Ken Palmer

7. Hyperbolic Sets of Ordinary Differential Equations

Abstract

In this chapter we develop the theory of hyperbolic sets for flows. First we show that the continuity of the splitting into stable and unstable bundles follows from the other items in the definition. Next we develop the theory of exponential dichotomies for linear differential equations, paying special attention to the roughness theorem. We use the latter to prove that hyperbolic sets are expansive both in a “continuous” way and a “discrete” way. Finally we show that hyperbolic sets are robust under perturbation, our major tool here being Lemma 2.17.

Ken Palmer

8. Transversal Homoclinic Orbits and Hyperbolic Sets in Differential Equations

Abstract

In this chapter, we show how to construct a hyperbolic set somewhat more complicated than a single hyperbolic periodic orbit. In Chapter 10 we shall use shadowing to show that the dynamics in the neighbourhood of this hyperbolic set is chaotic.

Ken Palmer

9. Shadowing Theorems for Hyperbolic Sets of Differential Equations

Abstract

For differential equations, it is not immediately clear how the shadowing theorem should be formulated and, in particular, how pseudo orbits should be defined. We consider two possibilities: first a discrete pseudo orbit such as would be obtained by numerically computing the solutions of a differential equation and then a continuous pseudo orbit which is needed for theoretical purposes. Shadowing theorems are proved for both kinds of pseudo orbits. We also study both discrete and continuous versions of expansivity. Then shadowing is used to show that there is a topological conjugacy between the flow on an isolated hyperbolic set and the flow on the nearby hyperbolic set for a perturbed flow. Also we use shadowing to show that isolated hyperbolic sets have the asymptotic phase property.

Ken Palmer

10. Symbolic Dynamics Near a Transversal Homoclinic Orbit of a System of Ordinary Differential Equations

Abstract

We consider the autonomous system of ordinary differential equations

$$\mathop x\limits^.= F(x),$$

(1)

where F : U → ℝⁿ is a C ¹ vectorfield, the set U being open and convex. Denote by ∅ the corresponding flow. Suppose u (t) is a hyperbolic periodic orbit for Eq.(l) with minimal period T < 0 and let p ₀ be in the intersection of the stable manifold W^s(u) and the unstable manifold W ^u(u). If p ₀ satisfies the transversality condition

$${T_{{p_0}}}{W^s}(u) \cap {T_{{p_0}}}{W^u}(u) = span\{ F({p_0})\} ,$$

then we know from Theorem 8.2 that the set

$$S = \{ u(t): - \infty< t < \infty \}\cup \{ {\phi ^t}({p_0}): - \infty< t < \infty \} $$

is hyperbolic. Now we state our main theorem, which uses symbolic dynamics to describe the solutions of Eq.(l) which remain in a neighbourhood of the set S.

Ken Palmer

11. Numerical Shadowing

Abstract

In this chapter our object is to demonstrate how shadowing ideas can be used to verify the accuracy of numerical simulations of dynamical systems and also how they can be used to rigorously establish the existence of periodic orbits and of chaotic behaviour.

Ken Palmer

Backmatter

Titel: Shadowing in Dynamical Systems
verfasst von: Ken Palmer
Verlag: Springer US
Electronic ISBN: 978-1-4757-3210-8
Print ISBN: 978-1-4419-4827-4
DOI: https://doi.org/10.1007/978-1-4757-3210-8

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Hyperbolic Fixed Points of Diffeomorphisms and Their Stable and Unstable Manifolds

2. Hyperbolic Sets of Diffeomorphisms

3. Transversal Homoclinic Points of Diffeomorphisms and Hyperbolic Sets

4. The Shadowing Theorem for Hyperbolic Sets of Diffeomorphisms

5. Symbolic Dynamics Near a Transversal Homoclinic Point of a Diffeomorphism

6. Hyperbolic Periodic Orbits of Ordinary Differential Equations, Stable and Unstable Manifolds and Asymptotic Phase

7. Hyperbolic Sets of Ordinary Differential Equations

8. Transversal Homoclinic Orbits and Hyperbolic Sets in Differential Equations

9. Shadowing Theorems for Hyperbolic Sets of Differential Equations

10. Symbolic Dynamics Near a Transversal Homoclinic Orbit of a System of Ordinary Differential Equations

11. Numerical Shadowing

Backmatter