On the effect of invisibility of stable periodic orbits at homoclinic bifurcations

S. V. Gonchenko, I. I. Ovsyannikov, D. Turaev

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


We study bifurcations of a homoclinic tangency to a saddle-focus periodic point. We show that the stability domain for single-round periodic orbits which bifurcate near the homoclinic tangency has a characteristic "comb- like" structure and depends strongly on the saddle value, i.e. on the area-contracting properties of the map at the saddle-focus. In particular, when the map contracts two-dimensional areas, we have a cascade of periodic sinks in any one-parameter family transverse to the bifurcation surface that corresponds to the homoclinic tangency. However, when the area-contraction property is broken (while three-dimensional volumes are still contracted), the cascade of single-round sinks appears with "probability zero" only. Thus, if three-dimensional volumes are contracted, chaos associated with a homoclinic tangency to a saddle-focus is always accompanied by stability windows; however the violation of the area-contraction property can make the stability windows invisible in one-parameter families.

Original languageEnglish
Pages (from-to)1115-1122
Number of pages8
JournalPhysica D: Nonlinear Phenomena
Issue number13
StatePublished - 1 Jul 2012
Externally publishedYes


  • Dynamical chaos
  • Henon map
  • Homoclinic tangency
  • Newhouse phenomenon

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


Dive into the research topics of 'On the effect of invisibility of stable periodic orbits at homoclinic bifurcations'. Together they form a unique fingerprint.

Cite this