On bifurcations of Lorenz attractors in the Lyubimov-Zaks model

We provide numerical evidence for the existence of the Lorenz and the Rovella (contracting Lorenz) attractors in the generalization of the Lorenz model proposed by Lyubimov and Zaks. The Lorenz attractor is robustly chaotic (pseudohyperbolic) in contrast to the Rovella attractor, which is only measure-persistent (it exists for a set of parameter values, which is nowhere dense but has a positive Lebesgue measure). It is well known that in this model, for certain values of parameters, there exists a homoclinic butterfly (a pair of homoclinic loops) to the symmetric saddle equilibrium, which is neutral, i.e., its eigenvalues λ₂<λ₁<0<γ are such that the saddle index ν=-λ₁/γ is equal to ∼1. The birth of the Lorenz attractor at this codimension-two bifurcation is established by means of numerical verification of the Shilnikov criterion. For the birth of the Rovella attractor, we propose a new criterion, which is also verified numerically.