Urs Kirchgraber

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.

A problem of orbital dynamics, which is separable in KS-variables

The motion of a rocket (with constant thrust) in the gravitational field of a mass point is studied. As is well known, the two dimensional version of this problem is separable in Levi-Civitavariables. It is shown here that, for the three dimensional case, the KS-variables produce the separability. A closed solution is found by using elliptic functions. Equivalent problems are mentioned.

Possible chaotic motion of comets in the Sun-Jupiter system: a computer-assisted approach based on shadowing

Using the shadowing technique we give an alternative computer-assisted proof of certain results in the restricted three-body problem made plausible by Koon et al (2000 Chaos 10 427–69), and rigorously established by Wilczak and Zgliczynski (2003 Commun. Math. Phys. 234 37–75) using topological horseshoes. The approach presented here seems to be rather efficient.

Verification of chaotic behaviour in the planar restricted three body problem

Abstract We present a computer-assisted method to verify the existence of chaotic behaviour in discrete dynamical systems. Our approach is closely related to the procedure introduced by Stoffer and Palmer [Nonlinearity 12 (1999) 1683]. Yet the scheme of Stoffer and Palmer is substantially improved leading to a much more efficient procedure with a wider range of potential applications. Moreover, the application to the planar restricted three body problem with two primaries of equal masses is offered. As is well known, this problem is described by a four-dimensional system of differential equations. Restricting the flow to an energy surface and defining an appropriate Poincare section the problem is reduced to a map in R 2 to which our method is shown to apply. Strictly speaking we do not rigorously establish chaotic behaviour. This is due to the fact that we replace certain rigorous error bounds by what we call realistic estimated upper bounds.

On the Stiefel-Baumgarte stabilization procedure

The purpose of this paper is to justify a stabilization procedure for ordinary differential equations due to Stiefel and Baumgarte. The main tools are methods and results from the theory of invariant manifolds of dynamical systems.ZusammenfassungIn dieser Arbeit wird ein von Stiefel und Baumgarte vorgeschlagenes Verfahren zur Stabilisierung von gewöhnlichen Differentialgleichungen theoretisch begründet. Als Hilfsmittel verwenden wir Methoden und Resultate aus der Theorie der invarianten Mannigfaltigkeiten dynamischer Systeme.

The transformational behaviour of perturbation theories

The transformational behaviour of Horis noncanonical perturbation theory (Hori 1971) as well as that of the theory of Krylov-Bogoliubof-Mitropolsky is studied. An integration procedure of the perturbation equations is based on the transformation properties that have been established.

On Some Invariant Manifold Results and Their Applications

A theory on the existence and the properties of invariant manifolds for a certain class of finite dimensional maps is described, with applications to averaging and to a problem in celestial mechanics.

On the Method of Averaging

In this paper we present several new aspects of the method of averages: first we describe some formal properties of the method, second we apply it in order to reprove Hopf’s bifurcation theorem (and obtain a direction of bifurcation formula which is similar to that of Hsu and Kazarinoff), thirdly we offer a theorem concerning error bounds. Since the details will appear elsewhere we only describe the results.

The structure of the secular system in nonlinear oscillations

If the method of averages is applied to a nearly integrable system of differential equations the so-called secular system is obtained. In the present paper we study the structure of the secular system for a large class of problems. We especially aim at establishing sufficient conditions for the secular system to be solvable by means of elementary functions.

Mathematik im Chaos

ZusammenfassungViele dynamische Vorgänge können wir unmittelbar wahrnehmen, zum Beispiel das Wetter, die Bewegung der Planeten, das Wirtschaftsgeschehen. Aus vielerlei Gründen besteht der Wunsch, über den Verlauf solcher Prozesse Prognosen zu gewinnen. Nun hat sich aber herausgestellt, daß es Bewegungsvorgänge gibt, die gar nicht vorhersagbar sind, obwohl sie durch ganz einfache Gesetze gesteuert werden! Dies hat zu einem gewissen Umbruch in unserem Naturverständnis geführt. Der Artikel beinhaltet eine Darstellung, die auch schon als Grundlage für einen zweitägigen Kurs mit Schülern der Sekundarstufe II verwendet wurde.