The simplest equivariant jerk system

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]

$\displaystyle \left\{ \begin{array}{l} \dot{x} = y \\ \dot{y} = z \\ \dot{z} = - \alpha z + x y^2 - x \\ \end{array} \right.$

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

$\gamma = \left[ \begin{array}{ccc} -1 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -1 \end{array} \right]$

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.

Chaotic attractor solution to the simplest equivariant jerk system

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.

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