The Jerk Dynamics of Lorenz Model

Jean-Marc Ginoux1, Riccardo Meucci2, Jaume Llibre3, Julien Clinton Sprott4
1Aix Marseille Univ, Universite de Toulon, CNRS, CPT, Marseille, France,
2National Institute of Optics - CNR, Florence, Italy,
3Departament de Matematiques, Universitat Autonoma de Barcelona, 08193 Bellaterra, Barcelona, Spain, and
4University of Wisconsin 1150 University Avenue Madison, WI 53706-1390 USA.

The Lorenz model is widely considered as the fi rst dynamical system exhibiting a chaotic attractor the shape of which is the famous butterfly. This similarity led Lorenz to name the sensitivity to initial conditions inherent to such chaotic systems, the butterfly effect making its model a paradigm of chaos. Nearly thirty years ago, Stefan J. Linz presented in a very interesting paper an "exact transformation" enabling to obtain the jerk form of the Lorenz model and a nonlinear transformation "simplifying its jerky dynamics". Unfortunately, the third order nonlinear differential equation he finally obtained precluded any mathematical analysis and made difficult numerical investigations since it contained exponential functions. In this work, we provide in the simplest way the jerk form of the Lorenz model. Then, a stability analysis of the jerk dynamics of Lorenz model prove that fixed points and their stability, eigenvalues, Lyapunov Characteristics Exponents and of course attractor shape are the exactly the same as those of Lorenz original model.

Ref: J. -M. Ginous, R. Meucci, J. Libre, and J. C. Sprott, Proceedings of the Third International Nonlinear Dynamics Conference (2023)

The complete paper is available in PDF format.

Return to Sprott's Books and Publications.