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

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.