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, ginoux@univ-tln.fr
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)