Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Homoclinic tangencies to resonant saddles and discrete Lorenz attractors

Pages: 273 - 288, Volume 10, Issue 2, April 2017      doi:10.3934/dcdss.2017013

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Sergey Gonchenko - Lobachevsky State University of Nizhny Novgorod, 23 Gagarin av., Nizhny Novgorod, 603950, Russian Federation (email)
Ivan Ovsyannikov - Universität Bremen, Bibliothekstrasse 1, 28359 Bremen, Germany (email)

Abstract: We study bifurcations of periodic orbits in three parameter general unfoldings of certain types quadratic homoclinic tangencies to saddle fixed points. We apply the rescaling technique to first return (Poincaré) maps and show that the rescaled maps can be brought to a map asymptotically close to the 3D Henon map $\bar x=y,\bar y=z,\bar z = M_1 + M_2 y + B x - z^2$ which, as known [14], exhibits discrete Lorenz attractors in some open domains of the parameters. Based on this, we prove the existence of infinite cascades of systems possessing discrete Lorenz attractors near the original diffeomorphism.

Keywords:  Homoclinic tangency, rescaling, 3D Hénon map, Poincare map, Lorenz attractor.
Mathematics Subject Classification:  Primary: 37C29, 37C70, 37G25.

Received: October 2015;      Revised: November 2016;      Available Online: January 2017.