Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2005-10-24
Reg. & Chaotic Dyn. 11(2) 191-212 (2006)
Nonlinear Sciences
Chaotic Dynamics
laTeX, 25 pages, 6 eps figures
Scientific paper
10.1070/RD2006v011n02ABEH000345
We study bifurcations of a three-dimensional diffeomorphism, $g_0$, that has a quadratic homoclinic tangency to a saddle-focus fixed point with multipliers $(\lambda e^{i\vphi}, \lambda e^{-i\vphi}, \gamma)$, where $0<\lambda<1<|\gamma|$ and $|\lambda^2\gamma|=1$. We show that in a three-parameter family, $g_{\eps}$, of diffeomorphisms close to $g_0$, there exist infinitely many open regions near $\eps =0$ where the corresponding normal form of the first return map to a neighborhood of a homoclinic point is a three-dimensional H\'enon-like map. This map possesses, in some parameter regions, a "wild-hyperbolic" Lorenz-type strange attractor. Thus, we show that this homoclinic bifurcation leads to a strange attractor. We also discuss the place that these three-dimensional H\'enon maps occupy in the class of quadratic volume-preserving diffeomorphisms.
Gonchenko S. V.
Meiss James D.
Ovsyannikov I. I.
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