Bifurcation to infinitely many sinks

Clark Robinson*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

85 Scopus citations

Abstract

This paper considers one parameter families of diffeomorphisms {Ft} in two dimensions which have a curve of dissipative saddle periodic points Pt, i.e. Ftn(Pt)=Pt and |det DFtn(Pt)|<1. The family is also assumed to create new homoclinic intersections of the stable and unstable manifolds of Pt as the parameter varies through t0. Gavirlov and Silnikov proved that if the new homoclinic intersections are created nondegenerately at t0, then there is an infinite cascade of periodic sinks, i.e. there are parameter values tn accumulating at t0 for which there is a sink of period n [GS2, Sect. 4]. We show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately. We give computer evidence to show that this latter result is probably applicable to the Hénon map for A near 1.392 and B equal -0.3. Newhouse proved a related result which showed the existence of infinitely many periodic sinks for a single diffeomorphism which is a perturbation of a diffeomorphism with a nondegenerate homoclinic tangency. We give the main geometric ideas of the proof of this theorem. We also give a variation of a key lemma to show that the result is true for a fixed one parameter family which creates a nondegenerate tangency. Thus under the nondegeneracy assumption, not only is there a cascade of sinks proved by Gavrilov and Silnikov, but also a single parameter value t* with infinitely many sinks.

Original languageEnglish (US)
Pages (from-to)433-459
Number of pages27
JournalCommunications in Mathematical Physics
Volume90
Issue number3
DOIs
StatePublished - Sep 1 1983

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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