Daniel Stoffer

Transversal homoclinic points and hyperbolic sets for non-autonomous maps I

A concept of generalized hyperbolic sets for non-autonomous maps is developed. Starting from transversal homoclinic orbits such generalized hyperbolic sets are constructed. The Shadowing Lemma is proven for maps admitting a generalized hyperbolic set. Time dependent symbolic dynamics is introduced and related to non-autonomous maps.ZusammenfassungDas Konzept von verallgemeinerten hyperbolischen Mengen für nicht-autonome Abbildungen wird entwickelt. Ausgehend von transversalen homoklinen Bahnen werden solche verallgemeinerte hyperbolische Mengen konstruiert. Das Shadowing Lemma wird für Abbildungen bewiesen, welche eine verallgemeinerte hyperbolische Menge haben. Es wird zeitabhängige symbolische Dynamik eingeführt und der Zusammenhang mit nicht-autonomen Abbildungen dargestellt.

Chaotic behaviour in simple dynamical systems

In this paper a description is given of the chaotic behaviour generated by a transversal homoclinic point of a plane map. A proof of Smale’s theorem via the shadowing property of hyperbolic sets is provided. The result is related to certain plane periodic systems of ODE’S like the periodically perturbed pendulum equation. To this end the so-called method of Melnikov is derived.

Invariant curves for variable step size integrators

AbstractThe behaviour of one-step methods with variable step size applied to

General linear methods: connection to one step methods and invariant curves

\dot x = f(x)

On the Application of Invariant Manifold Theory, in particular to Numerical Analysis

is investigated. Usually the step size for the current step depends on one or several previous steps. However, under some natural assumptions it can be shown that the step size asymptotically depends only on the locationx. This allows to introduce anx-dependent time transformation taking a variable step size method to a constant step-size method. By means of such a transformation general properties of constant step size methods carry over to variable step size methods. This is used to show that if the differential equation admits a hyperbolic periodic solution the variable step size method admits an invariant closed curve near the orbit of the periodic solution.

Runge-Kutta solutions of stiff differential equations near stationary points

AbstractLet

Two results on stable rapidly oscillating periodic solutions of delay differential equations

\dot x = f(t,x)