Preprint des Projektes: DFG Forschergruppe Spektrale Analysis, asymptotische Verteilungen und stochastische Prozesse
Numerische Approximation und Spektrale Analysis unendlich-dimensionaler Dynamischer Systeme
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.