Abstract
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 language | English |
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Pages (from-to) | 1115-1122 |
Number of pages | 8 |
Journal | Physica D: Nonlinear Phenomena |
Volume | 241 |
Issue number | 13 |
DOIs | |
State | Published - 1 Jul 2012 |
Externally published | Yes |
Keywords
- Dynamical chaos
- Henon map
- Homoclinic tangency
- Newhouse phenomenon
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics