Discretizations of dynamical systems with a saddle-node homoclinic orbit

W.-J. Beyn; Y.-K Zou

doi:10.3934/dcds.1996.2.351

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1996, Volume 2, Issue 3: 351-365. Doi: 10.3934/dcds.1996.2.351

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Discretizations of dynamical systems with a saddle-node homoclinic orbit

W.-J. Beyn^1, and
Y.-K Zou^2,

1.
Fakultät Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld
2.
Department of Mathematics, Jilin University, Changchun 130023

Received: February 1996

Published: July 1996

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Abstract

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 [1], [5] on homoclinic discretization effects in the hyperbolic case.

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Mathematics Subject Classification: Primary: 65L12; Secondary: 58F30, 58F08, 34C45.

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