Preprint 5/96 in the DFG-Schwerpunktprogramm "Ergodentheorie, Analysis und effiziente Simulation dynamischer Systeme":

Y.-K. Zou, W.-J. Beyn: Discretizations of dynamical systems with a saddle-node homoclinic orbit

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.

preprint_5_96.ps (837KB, includes 5 illustrations)

Other preprints in the DFG-Project Connecting Orbits.

DFG-Project Connecting Orbits (Bielefeld)

DFG-Schwerpunktprogramm Dynamik (Berlin)

Department of Mathematics | University of Bielefeld