The Jerk Dynamics of Lorenz Model

The Lorenz model is widely considered as the first 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 30 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 proves that fixed points and their stability, eigenvalues, Lyapunov characteristic exponents, and of course attractor shape are exactly the same as those of the original Lorenz model.