Multi-winged Lorenz attractors due to bifurcations of a periodic orbit with multipliers $(-1,i,-i)$

TL;DR

This paper demonstrates how specific bifurcations of periodic orbits in a quadratic Hénon map lead to the emergence of robust, Lorenz-like chaotic attractors, including new multi-winged variants called Simo angels.

Contribution

It introduces new types of Lorenz-like attractors arising from bifurcations with multipliers (-1,i,-i) in a quadratic Hénon map, supported by numerical analysis of a symmetric normal form.

Findings

Existence of three types of Lorenz-like attractors in the Hénon map.

Identification of parameter regions with pseudohyperbolic, robustly chaotic attractors.

Demonstration that all orbits in these attractors have positive Lyapunov exponents.

Abstract

We show that bifurcations of periodic orbits with multipliers $(-1,i,-i)$ can lead to the birth of pseudohyperbolic (i.e., robustly chaotic) Lorenz-like attractors of three different types: one is a discrete analogue of the classical Lorenz attractor, and the other two are new. We call them two- and four-winged ``Sim\'o angels''. These three attractors exist in an orientation-reversing, three-dimensional, quadratic H\'enon map. Our analysis is based on a numerical study of a normal form for this bifurcation, a three-dimensional system of differential equations with a Z4-symmetry. We investigate bifurcations in the normal form and describe those responsible for the emergence of the Lorenz attractor and the continuous-time version of the Simo angels. Both for the normal form and the 3D H\'enon map, we have found open regions in the parameter space where the attractors are pseudohyperbolic, implying that for every parameter value from these regions every orbit in the attractor has positive top Lyapunov exponent.