The simplest equivariant jerk system

 The system

A jerk system is a nonlinear dynamical system which can be rewritten under a canonical form, that is, as a function of one of its dynamical variables and its successive derivatives. Jean-Marc Malasoma proposed the simplest equivariant jerk system reading as [1]

where \(y=\dot{x}\) and \(z=\ddot{x}\). This system is equivariant, that is, it obeys to the relation
\(\gamma \cdot \mb{f} (\mb{x}) = \mb{f} (\gamma \cdot \mb{x})\)
where \(\gamma\) is a \(3 \times 3\) matrix defining the symmetry properties. In
the present case, the \(\gamma\)-matrix

defines an inversion symmetry \({\cal P}\). It means that the vector field \(\mb{f}\) is invariant when \((x,y,z)\) are mapped into \((-x,-y,-z)\). This simplest equivariant jerk system has a single fixed point \(F_0\) located at the origin of the phase space. It is a saddle-focus with one negative real eigenvalue and two complex conjugate eigenvalues with positive real part. With \(\alpha = 2.027717\), a chaotic attractor is obtained.

This system was investigated in terms of its image, that is, under the two-to-one mapping allowing to obtain a projection of the dynamics without any residual symmetry [2]. The inversion symmetry of the simplest equivariant system is therefore modded out. The bifurcation diagram can be thus predicted from the unimodal order although the first-return map computed in the original phase space exhibits three critical points. This feature is the same than the one observed on the Burke & Shaw system although this latter system has a rotation symmetry.