Collision, explosion and collapse of homoclinic classes
Abstract
Homoclinic classes of generic C^{1}diffeomorphisms are maximal transitive sets and pairwise disjoint. We present here a model explaining how two different homoclinic classes may intersect, failing to be disjoint. For that we construct a oneparameter family of diffeomorphisms (g_{s})_{sepsi[1,1]} with hyperbolic points P and Q having nontrivial homoclinic classes such that for s < 0 the classes of P and Q are disjoint; for s = 0 the classes collide and their intersection is a saddlenode and for s > 0, after an explosion, the two classes are equal. Our constructions involve bifurcations through heterodimensional and saddlenode cycles.
This paper was partially supported by CAPES, CNPq, Faperj and Pronex Dynamical Systems (Brazil).
 Publication:

Nonlinearity
 Pub Date:
 May 2004
 DOI:
 10.1088/09517715/17/3/013
 arXiv:
 arXiv:1205.1023
 Bibcode:
 2004Nonli..17.1001D
 Keywords:

 Mathematics  Dynamical Systems;
 37G25 (Primary) 37D30;
 37E99;
 37G30 (Secondary)
 EPrint:
 This is the final version, accepted in 2004