Six Lectures on Dynamical Systems

The following sections are included:

• Preface

• Contents

Of importance for the study of smooth dynamical systems, the concepts of Lyapunov function, attractor and chain recurrence are entirely topological. The theory connecting these ideas provides tools with which to begin the study of a wide variety of dynamical systems.

This lecture is designed to meet two seemingly contrary purposes. On one hand it is believed to enable a novice to approach the theory of invariant and integral manifolds through a completely selfcontained presentation, and on the other hand it is thought that even the experts may gain some new insight into the theory of integral manifolds, mainly due to a new approach allowing a generalization of the local invariant manifold theory to nonautonomous differential equations ẋ = f(t,x) in Banach spaces with possibly discontinuous t-dependence. More detailed information is given in Section 1.

Control systems can be considered as dynamical systems on the space , where is the set of admissible control functions, and the manifold M is the state space of the control system. This point of view allows us to study in this survey paper single trajectories depending on the control and the initial value, and it results in a number of connections between control theory and the theory of dynamical systems. On the one hand, one obtains new results for control theoretic questions, on the other hand one finds new (explicit) examples for many concepts in dynamical systems. Of particular importance are concepts like topological transitivity, chain recurrence, limit sets, Morse decompositions, linearized flows, subbundle decompositions and spectral theory (Lyapunov exponents), and their control theoretic analogues. This approach also yields, among others, new perturbation theorems for ordinary differential equations. We will study in detail applications to the (feedback) control about hyperbolic fixed points and the controllability around the attractor in the Lorenz equations (‘control of chaos’).

In this paper three main aspects of shadowing in discrete dynamical systems are addressed. First, a new proof of the classical infinite-time shadowing theorem for hyperbolic diffeomorphisms is given. Second, a finite-time shadowing theorem for maps which do not satisfy the hyperbolicity requirement is presented. The use of this theorem in numerical computations is demonstrated. Finally, a new theory of periodic shadowing for maps is developed. This theory allows one to verify the existence of periodic orbits, including long and unstable ones, and estimate their Lyapunov exponents from numerical computations.

The lecture presents the results on the perturbation problem of invariant manifolds of smooth dynamical systems given by a general autonomous ordinary differential equation in Rⁿ. The submitted results cover the problem of perturbation of equilibrium points, periodic orbits, locally invariant manifolds, normally hyperbolic compact invariant manifolds and compact invariant manifolds which are not locally unique. Various phenomena occurring under a perturbation of invariant manifolds are illustrated by examples.

In an arbitrary complete metric space discrete dynamical (semidynamical) systems generated by homeomorphisms (continuous mappings) are considered. Necessary and sufficient conditions for the existence of invariant sets are obtained. In this context the problem of decoupling and simplifying dynamical and semidynamical systems by means of topological transformations are studied. The results obtained allow to reduce the investigation of the given system to an analogous investigation for a much simpler system.

In consequence, the classical Grobman–Hartman theorem as well as the principle of reduction for semilinear dynamical systems in Euclidean and Banach spaces are obtained.