A Lorenz-like system with an inversion symmetry: Topology of some of its attractors

In a study devoted to discrete Lorenz systems, Gonchenko and co-workers [Chaos 32, 121107 (2022)] briefly mentioned a three-dimensional system that would be a Lorenz-like system if a quadratic monomial was not replaced with a cubic one. Such a modification introduces an inversion symmetry, rendering this system atypical and whose dynamics deserves to be investigated. This is here performed along a line of the parameter space in terms of the template for some of its attractors. There are similarities with the classical bifurcation diagram produced by the Lorenz system with R as the bifurcation parameter and some differences that are exhibited. In particular, it is shown that an attractor merging crisis with the saddle point located at the origin of the state space allows us to produce an attractor that is similar to the one observed in the Moore-Spiegel system long ago [Letellier, Ph.D. thesis, Paris vii University (1994)] and that is rarely observed in Lorenz-like systems. We also took the opportunity to show how it may be useful to investigate the structure of the dynamics that can be safely characterized with a first-return map built on {Δτn=τn-τn-1}, where τn is the duration of the return to a surface of section.