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

被引:9
|
作者
Gonchenko, S. V. [2 ]
Ovsyannikov, I. I. [3 ]
Turaev, D. [1 ]
机构
[1] Univ London Imperial Coll Sci Technol & Med, Dept Math, London SW7 2AZ, England
[2] Nizhnii Novgorod State Univ, Inst Appl Math & Cybernet, Nizhnii Novgorod 603005, Russia
[3] Nizhnii Novgorod State Univ, Dept Radiophys, Nizhnii Novgorod 603950, Russia
关键词
Homoclinic tangency; Dynamical chaos; Henon map; Newhouse phenomenon; DIFFEOMORPHISMS; SYSTEMS; TANGENCIES;
D O I
10.1016/j.physd.2012.03.002
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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. (C) 2012 Elsevier By. All rights reserved.
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页码:1115 / 1122
页数:8
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