Homoclinic tangencies to resonant saddles and discrete Lorenz attractors

TL;DR

This paper investigates bifurcations of periodic orbits near quadratic homoclinic tangencies to saddle points, demonstrating the emergence of Lorenz-like attractors through rescaling techniques applied to Poincaré maps.

Contribution

It introduces a method to relate bifurcations near homoclinic tangencies to the 3D Henon map, proving the existence of infinite cascades of Lorenz-like attractors.

Findings

Rescaled Poincaré maps approximate the 3D Henon map.

Existence of infinite cascades of Lorenz-like attractors.

Identification of parameter domains with wild hyperbolic attractors.

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 (Poincar\'e) 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, exhibits wild hyperbolic Lorenz-like attractors in some open domains of the parameters. Based on this, we prove the existence of infinite cascades of Lorenz-like attractors.