Center for Applied Mathematics
657 Frank H.T. Rhodes Hall, Cornell University
Ithaca, NY 14853, USA
email: ck274@cornell.edu
Christian Kuehn
Abstract:
Geometric Singular Perturbation Theory (GSPT) and
Conley Index Theory are two powerful techniques to analyze
dynamical systems. Conley already realized that using his index
is easier for singular perturbation problems. In this paper,
we will revisit Conley's results and prove that the GSPT technique
of Fenichel Normal Form can be used to simplify the application
of Conley index techniques even further. We also hope that our
results provide a better bridge between the different fields.
Furthermore we show how to interpret Conley's conditions in
terms of averaging. The result are illustrated by the two-dimensional
van der Pol equation and by a three-dimensional Morris-Lecar model.
Submitted May 19, 2010. Published August 4, 2010.
Math Subject Classifications: 34C26, 34E15, 37B30, 34A26, 34C45.
Key Words: Fast-slow system; Conley index; Fenichel normal form;
transversality.
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Christian Kuehn Center for Applied Mathematics 657 Frank H.T. Rhodes Hall, Cornell University Ithaca, NY 14853, USA email: ck274@cornell.edu |
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