Basins for general nonlinear hénon attracting sets

Sam Patterson*, Clark Robinson

*Corresponding author for this work

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

When a new homoclinic intersection is created for a dissipative diffeomorphism in dimension two, there results a cascade of sinks. We show that immediately after one of these sinks qn is formed, its basin boundary is made up of the stable manifold of the saddle periodic point pn formed at the same time. After this sink undergoes a period doubling, there still remains a trapping region with an attracting set inside. In fact, we show that until this saddle periodic point pn has its own homoclinic bifurcation, there is an attracting set whose boundary is made up of the stable manifold of pn- By picking a rectangle Bn carefully, the one-parameter family of maps ftn creates these sinks and attracting sets by pulling the image ftn(Bn) across Bn and eventually forming a horseshoe in Bn. The maps, ftn on Bn, are well approximated for large n by quadratic maps equivalent to the Hénon map. We prove our results for general nonlinear Hénon maps which include not only the quadratic maps but also other nonlinear maps which also create horseshoes, including those arising from homoclinic tangencies.

Original languageEnglish (US)
Pages (from-to)615-623
Number of pages9
JournalProceedings of the American Mathematical Society
Volume103
Issue number2
DOIs
StatePublished - Jun 1988

Keywords

  • Basins
  • Creation of horseshoes
  • Homoclinic tangencies
  • Sinks

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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