Ken Palmer

Rigorous verification of chaotic behaviour of maps using validated shadowing

In this paper discrete dynamical systems exhibiting `complicated behaviour are investigated. We present a computer-assisted method to prove that the given system admits the shift map as a subsystem. The method is applied to the Henon map with the classical parameter values.

Long periodic shadowing

A general theorem for establishing the existence of a true periodic orbit near a numerically computed pseudoperiodic orbit of an autonomous system of ordinary differential equations is presented. For practical applications, a Newton method is devised to compute appropriate pseudoperiodic orbits. Then numerical considerations for checking the hypotheses of the theorem in terms of quantities which can be computed directly from the pseudoperiodic orbit and the vector field are addressed. Finally, a numerical method for estimating the Lyapunov exponents of the true periodic orbit is given. The theory and computations are designed to be applicable for unstable periodic orbits with long periods. The existence of several such periodic orbits of the Lorenz equations is exhibited.

Shadowing orbits of ordinary differential equations

Abstract A new notion of shadowing of a pseudo orbit, an approximate solution, of an autonomous system of ordinary differential equations by an associated nearby true orbit is introduced. Then a general shadowing theorem for finite time, which guarantees the existence of shadowing in ordinary differential equations and provides error bounds for the distance between the true and the pseudo orbit in terms of computable quantities, is proved. The use of this theorem in numerical computations of orbits is outlined.

Transversal connecting orbits from shadowing

A rigorous numerical method for establishing the existence of a transversal connecting orbit from one hyperbolic periodic orbit to another of a differential equation in

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

{mathbb{R}^n}

Hyperbolic Fixed Points of Diffeomorphisms and Their Stable and Unstable Manifolds

is presented. As the first component of this method, a general shadowing theorem that guarantees the existence of such a connecting orbit near a suitable pseudo connection orbit given the invertibility of a certain operator is proved. The second component consists of a refinement procedure for numerically computing a pseudo connecting orbit between two pseudo periodic orbits with sufficiently small local errors so as to satisfy the hypothesis of the theorem. The third component consists of a numerical procedure to verify the invertibility of the operator and obtain a rigorous upper bound for the norm of its inverse. Using this method, existence of chaos is demonstrated on examples with transversal homoclinic orbits, and with cycles of transversal heteroclinic orbits.

Transversal Homoclinic Orbits and Hyperbolic Sets in Differential Equations

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 Poincare map. However, first we begin by recalling a few elementary facts from the theory of ordinary differential equations.

The Shadowing Theorem for Hyperbolic Sets of Diffeomorphisms

We consider the autonomous system of ordinary differential equations n n