Vladimir Gonchenko

10, Ulyanova st. 603005, Nizhny Novgorod, Russia
Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University


Gonchenko S. V., Gonchenko V. S., Shilnikov L. P.
On a homoclinic origin of Hénon-like maps
2010, vol. 15, nos. 4-5, pp.  462-481
We review bifurcations of homoclinic tangencies leading to Hénon-like maps of various kinds.
Keywords: homoclinic tangency, Hénon-like maps, saddle-focus fixed point, wild-hyperbolic attractor
Citation: Gonchenko S. V., Gonchenko V. S., Shilnikov L. P.,  On a homoclinic origin of Hénon-like maps, Regular and Chaotic Dynamics, 2010, vol. 15, nos. 4-5, pp. 462-481
Gonchenko S. V., Gonchenko V. S., Tatjer J. C.
We study bifurcations of periodic orbits in two parameter general unfoldings of a certain type homoclinic tangency (called a generalized homoclinic tangency) to a saddle fixed point. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to so-called generalized Hénon maps which have non-degenerate bifurcations of fixed points including those with multipliers $e^{\pm i \phi}$. On the basis of this, we prove the existence of infinite cascades of periodic sinks and periodic stable invariant circles.
Keywords: homoclinic tangency, rescaling, generalized Henon map, bifurcation
Citation: Gonchenko S. V., Gonchenko V. S., Tatjer J. C.,  Bifurcations of Three-Dimensional Diffeomorphisms with Non-Simple Quadratic Homoclinic Tangencies and Generalized Hénon Maps, Regular and Chaotic Dynamics, 2007, vol. 12, no. 3, pp. 233-266

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