Jean-Marc MALASOMA
29/01/2008

**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 and . This system is equivariant, that is, it obeys to the relation where is a matrix defining the symmetry properties. In the present case, the -matrix

defines an inversion symmetry . It means that the vector field is invariant when are mapped into . This simplest equivariant jerk system has a single fixed point 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 , 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.

[1] **J.-M. Malasoma**, What is the simplest dissipative chaotic jerk equation which is parity invariant?, *Physics Letters A*, **264**, 383-389, 2000.

[2] **C. Letellier & J.-M. Malasoma**,
Unimodal order in the image of the simplest equivariant jerk system,
*Physical Review E*, **64**, 067202, 2001.