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BIT Numerical Mathematics

, Volume 56, Issue 2, pp 605–631 | Cite as

Nonautonomous systems with transversal homoclinic structures under discretization

  • Alina Girod
  • Thorsten Hüls
Article

Abstract

We consider homoclinic orbits in continuous time nonautonomous dynamical systems. Unlike the autonomous case, stable and unstable fiber bundles that generalize stable and unstable manifolds, typically intersect transversally in isolated points. In the first part, we establish persistence and error estimates for one-step discretizations of transversal homoclinic orbits. Secondly, we extend an algorithm by England, Krauskopf, Osinga to nonautonomous systems and illustrate transversally intersecting fibers along homoclinic orbits for three examples. The first one is constructed artificially in order to study numerical errors, while the second one is a periodically forced model that reveals the influence of underlying autonomous dynamics. The third example originates from mathematical biology.

Keywords

Nonautonomous dynamical systems Homoclinic orbits  Fiber bundles Discretization effects Approximation theory Exponential dichotomy 

Mathematics Subject Classification

70K44 55R10 37B55 34C37 

Notes

Acknowledgments

The authors thank an anonymous referee for constructive and helpful remarks, which improved the first version of this paper.

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsBielefeld UniversityBielefeldGermany

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