Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps

Sergey Gonchenko, Dmitry Turaev, Leonid Shilnikov

Research output: Contribution to journalArticlepeer-review

50 Scopus citations

Abstract

We show that maps with homoclinic tangencies of arbitrarily high orders and, as a consequence, with arbitrarily degenerate periodic orbits are dense in the Newhouse regions in spaces of real-analytic area-preserving two-dimensional maps and general real-analytic two-dimensional maps (the result was earlier known only for the space of smooth non-conservative maps). Based on this, we show that a generic area-preserving map from the Newhouse region is 'universal' in the sense that its iterations approximate the dynamics of any other area-preserving map with arbitrarily good accuracy. In fact, we show that every dynamical phenomenon which occurs generically in any open set of symplectic diffeomorphisms of a two-dimensional disc, or in any open set of finite-parameter families of such diffeomorphisms, can be encountered at a perturbation of any area-preserving two-dimensional map with a homoclinic tangency.

Original languageEnglish
Article number002
Pages (from-to)241-275
Number of pages35
JournalNonlinearity
Volume20
Issue number2
DOIs
StatePublished - 1 Feb 2007

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • General Physics and Astronomy
  • Applied Mathematics

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