2022 An atypical Lorenz-like system with an inversion symmetry

In a study devoted to discrete Lorenz systems, Gonchenko and coworkers [1] 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 in- version symmetry, rending this system atypical.

In 2022, Sergey Gonchenko and his collaborators proposed a Lorenz-like system with an inversion symmetry \(\Gamma_0\) combined with the common rotation symmetry \({\cal R}_z^\pi\) around the \(z\)-axis. The equations governing the dynamics of this system are

This system is equivariant under three different symmetries, each of them characterized by a \(3 \times 3\) matrix \(M\) leading to

where \(\mb{x} \in \mathbb{R}^3\) is the state vector and \(\mb{f}\) is the vector field [2] There is an inversion symmetry \(\Gamma_0 : \mathbb{R}^3 \rightarrow \mathbb{R}^3\) defined as

and a rotation symmetry \({\cal R}_z^\pi\) around the \(z\)-axis defined as

Both are order-2 symmetry, that is, \(\Gamma_0^2 = \mathbb{I}\) and \(({\cal R}_z^\pi)^2 = \mathbb{I}\) where \(\mathbb{I}\) is the identity matrix. In fact, this system is also equivariant under a reflection symmetry \(\sigma_x\) with respect to the \(x\)-\(y\) plane defined as

The normal axis to the \(x\)-\(y\) plane is used to designate this symmetry since it is the variable that is transformed.

The Gonchenko system produces a Lorenz attractor [Fig. 1], that is, an attractor topologically equivalent to the attractor produced by the Lorenz system with Lorenz’s parameter values when \(a = 0.087\) and \(b = 0.40\). There are two co-existing attractors, \({\cal A}_{\rm L}^+\) and \({\cal A}_{\rm L}^-\), that are globally left invariant under \({\cal R}_z^\pi\) and symmetry related under \(\Gamma_0\) or \(\sigma_z\). More about this system in [3]