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Tangencies and the Conley index

Published online by Cambridge University Press:  06 August 2002

ZIN ARAI
ZIN ARAI
Affiliation:
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan (e-mail: arai@kusm.kyoto-u.ac.jp)
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Abstract

We construct an algebraic method for detecting bifurcations of discrete dynamical systems using the Conley index theory. We define the transition matrix pair for discrete semidynamical systems, which detects bifurcations of connecting orbits. As an application, we give a sufficient condition for the occurrence of homoclinic and heteroclinic tangencies by showing that the occurrence of a heterodimensional cycle in the projectivization of a dynamical system implies the tangency of the original system.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 22 , Issue 4 , August 2002 , pp. 973 - 999
Copyright
© 2002 Cambridge University Press

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