Preprint des Projektes: DFG Forschergruppe Spektrale Analysis, asymptotische Verteilungen und stochastische Prozesse

Numerische Approximation und Spektrale Analysis unendlich-dimensionaler Dynamischer Systeme

03-008 Yong-Kui Zou, Wolf-Jürgen Beyn.
On the existence of transversal heteroclinic orbits in discretized dynamical systems

In this paper we prove the existence of transversal heteroclinic orbits for maps that are obtained from one-step methods applied to a continuous dynamical system. It is assumed that the continuous system exhibits a heteroclinic orbit at a specific value of a parameter. While it is known that analytic vector fields lead to exponentially small splittings of separatrices in the discrete system, we analyze here the case of a continuous system that is smooth of finite order only. Assuming that a certain derivative has a jump discontinuity at a specific hyperplane we show that the discretized systems have transversal heteroclinic orbits. The essential step in deriving such a result is a refinement of a previously developed error analysis which applies exponential dichotomy and Fredholm techniques to the discretized system.