Lorenzo J Díaz and Bianca Santoro 2004 Nonlinearity 17 1001 doi:10.1088/0951-7715/17/3/013
Collision, explosion and collapse of homoclinic classes*
Lorenzo J Díaz1 and Bianca Santoro2
Show affiliationsRecommended by L Bunimovich
Homoclinic classes of generic C1-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 one-parameter family of diffeomorphisms (gs)s
[−1,1] with hyperbolic points P and Q having non-trivial 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 saddle-node and for s > 0, after an explosion, the two classes are equal. Our constructions involve bifurcations through heterodimensional and saddle-node cycles.
- Footnote
- PACS
- MSC
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37D30 Partially hyperbolic systems and dominated splittings
37G20 Hyperbolic singular points with homoclinic trajectories
- Subjects
- Dates
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Issue 3 (May 2004)
Received 4 December 2002, in final form 30 January 2004
Published 8 March 2004
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Collision, explosion and collapse of homoclinic classes
Lorenzo J Díaz and Bianca Santoro 2004 Nonlinearity 17 1001
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