J. Tatjer

585, Barcelona 08007, Spain
Departament de Matematica Aplicada i Analisi, Gran Via de les Corts Catalanes


Gonchenko S. V., Ovsyannikov I. I., Tatjer J. C.
It was established in [1] that bifurcations of three-dimensional diffeomorphisms with a homoclinic tangency to a saddle-focus fixed point with the Jacobian equal to 1 can lead to Lorenz-like strange attractors. In the present paper we prove an analogous result for three-dimensional diffeomorphisms with a homoclinic tangency to a saddle fixed point with the Jacobian equal to 1, provided the quadratic homoclinic tangency under consideration is nonsimple.
Keywords: Homoclinic tangency, rescaling, 3D Hénon map, bifurcation, Lorenz-like attractor
Citation: Gonchenko S. V., Ovsyannikov I. I., Tatjer J. C.,  Birth of Discrete Lorenz Attractors at the Bifurcations of 3D Maps with Homoclinic Tangencies to Saddle Points, Regular and Chaotic Dynamics, 2014, vol. 19, no. 4, pp. 495-505
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

Back to the list