Preprint des Projektes: DFG Schwerpunkt Programm: Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme
Verbindungsorbits in hochdimensionalen dynamischen Systemen
We consider a parametrized dynamical system with a homoclinic orbit that connects the center manifold of a saddle node to its strongly stable manifold. This is a codimension 2 homoclinic bifurcation with a well known unfolding. We show that the map obtained by discretizing such a system with a one-step method (the centered Euler scheme), inherits a discrete saddle-node homoclinic orbit. This orbit occurs on the line of saddle nodes and, as the numerical results suggest, there is actually a closed curve of such orbits and almost all of them consist of transversal homoclinic points. Our results complement those of Beyn (1987), Fiedler and Scheurle (1991) on homoclinic discretization effects in the hyperbolic case.