Analytic proof of the emergence of new type of Lorenz-like attractors from the triple instability in systems with $\mathbb{Z}_4$-symmetry

TL;DR

This paper provides an analytical proof that systems with $\

Contribution

It introduces a new normal form for triple zero eigenvalue bifurcations in $\

Findings

Lorenz attractors can emerge from triple zero eigenvalue bifurcations.

Simo angels are also shown to arise from the same bifurcation.

The results confirm numerical observations of Lorenz-like attractors in 3D maps.

Abstract

We study bifurcations of a symmetric equilibrium state in systems of differential equations invariant with respect to a $\mathbb{Z}_4$-symmetry. We prove that if the equilibrium state has a triple zero eigenvalue, then pseudohyperbolic attractors of different types can arise as a result of the bifurcation. The first type is the classical Lorenz attractor and the second type is the so-called Sim\'o angel. The normal form of the considered bifurcation also serves as the normal form of the bifurcation of a periodic orbit with multipliers $(-1, i, -i)$. Therefore, the results of this paper can also be used to prove the emergence of discrete pseudohyperbolic attractors as a result of this codimension-3 bifurcation, providing a theoretical confirmation for the numerically observed Lorenz-like attractors and Sim\'o angels in the three-dimensional H\'enon map.