Analytically tractable reconstruction of singular hyperbolic and quasi-strange attractors of Lorenz-type systems

TL;DR

This paper analytically investigates bifurcations in a piecewise linear Lorenz-type system, revealing explicit forms of bifurcations and comparing them with smooth Lorenz models, to understand the emergence of strange and quasi-strange attractors.

Contribution

It provides explicit analytical descriptions of bifurcations leading to strange and quasi-strange attractors in a piecewise linear Lorenz-type system, extending previous work to negative saddle values.

Findings

Explicit forms of homoclinic and pitchfork bifurcations for negative saddle value.

Identification of a cascade leading to quasi-strange attractors.

Numerical validation with the Lorenz-Lyubimov-Zaks system.

Abstract

We consider a certain three-dimensional piecewise linear system of Lorenz type in the cases of positive and negative saddle value, which is the sum of two eigenvalues of the saddle nearest to zero. This system was recently proposed and studied in \cite{belykh2019lorenz} for the case of the positive saddle value. Here we consider the main bifurcations leading to the birth of the strange attractor and provide its comparison with those of original smooth Lorenz model. For the case of a negative saddle value, we obtain an explicit forms of homoclinic and pitchfork bifurcations forming a codimension 1 cascade leading to the emergence of a quasi-strange attractor. This cascade reproduces a typical bifurcation route towards the birth of a quasi-strange attractor in smooth Lorenz-like systems with the negative saddle value. We support our analytical result with a numerical comparison with the Lorenz-Lyubimov-Zaks system.