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Three-dimensional dissipative diffeomorphisms with homoclinic tangencies

Published online by Cambridge University Press:  26 March 2001

JOAN CARLES TATJER
Affiliation:
Dept. Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain

Abstract

Given a two-parameter family of three-dimensional diffeomorphisms \{f_{a,b}\}_{a,b}, with a dissipative (but not sectionally dissipative) saddle fixed point, assume that a special type of quadratic homoclinic tangency of the invariant manifolds exists. Then there is a return map f^n_{a,b}, near the homoclinic orbit, for values of the parameter near such a tangency and for n large enough, such that, after a change of variables and reparametrization depending on n, this return map tends to a simple quadratic map. This implies the existence of strange attractors and infinitely many sinks as in other known cases. Moreover, there appear attracting invariant circles, implying the existence of quasi-periodic behaviour near the homoclinic tangency.

Type
Research Article
Copyright
© 2001 Cambridge University Press

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