American Institute of Mathematical Sciences

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July  1996, 2(3): 351-365. doi: 10.3934/dcds.1996.2.351

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, Germany
2. 

Department of Mathematics, Jilin University, Changchun 130023, China

Received  February 1996 Published  May 1996

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.
Keywords: Homoclinic orbits, discrete homoclinic orbits., one-step methods, saddle-node.
Mathematics Subject Classification: Primary: 65L12; Secondary: 58F30, 58F08, 34C4.
Citation: W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
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