Homoclinic tangencies to resonant saddles and discrete Lorenz attractors

Sergey Gonchenko; Ivan Ovsyannikov

doi:10.3934/dcdss.2017013

Article Contents

2017, Volume 10, Issue 2: 273-288. Doi: 10.3934/dcdss.2017013

This issue Previous Article Random interval diffeomorphisms Next Article A family of self-avoiding random walks interpolating the loop-erased random walk and a self-avoiding walk on the Sierpiński gasket

Homoclinic tangencies to resonant saddles and discrete Lorenz attractors

Sergey Gonchenko^1, and
Ivan Ovsyannikov^1,2, ,

1.
Lobachevsky State University of Nizhny Novgorod, 23 Gagarin av., Nizhny Novgorod, 603950, Russia
2.
Universität Bremen, Bibliothekstrasse 1,28359 Bremen, Germany

* Corresponding author:Ivan Ovsyannikov.
* Corresponding author:Ivan Ovsyannikov.

Received: October 2015

Revised: November 2016

Published: February 2018

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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.

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Mathematics Subject Classification: Primary: 37C29, 37C70, 37G25.

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Figure 1. Schematic examples of pseudohyperbolic attractors. (a) Shilnikov-Turaev wild spiral attractor for a four-dimensional flow. (b) A discrete super-Lorenz attractor that contains a saddle fixed point $O$, pieces of its unstable (one-dimensional) manifolds are drown. Fixed points $O_1$ and $O_2$ (with $\dim W^u(O_i) = 2$ do not belong to the attractor, they are posed in "holes".

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Figure 2. Plots of attractors of map (1) observed numerically in [14] for $M_1=0,B=0.7$ and $M_2=0.85$ (left) or $M_2=0.815$ (right). In the left panel, the projection on the $(x,y)$-plane is also displayed. In the right panel, a "figure-eight" saddle closed invariant curve inside the lacuna is shown.

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Figure 3. The main steps of the creation of a discrete (c) Lorenz attractor; (d) figure-eight attractor.

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Figure 4. Two examples of 3D diffeomorphisms with nontransversal heteroclinic cycles containing two fixed points $O_1$ and $O_2$ of type $(2,1)$: (a) $O_1$ is a saddle-focus, this case was studied in [20]; (b) $O_1$ and $O_2$ are both saddle-foci, this case was studied in [13].

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Figure 5. Examples of 3D maps with quadratic homoclinic tangencies to a saddle fixed point $O$ of conservative type (whose bifurcations lead to the birth of discrete Lorenz attractors): (a) $O$ is a conservative saddle-focus [12]; (b) $O$ is a general saddle and the tangency is not simple [15]; (c) $O$ is a resonant saddle ($\lambda_1 = -\lambda_2$), studied in the present paper.

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