Chen's attractor exists if Lorenz repulsor exists: The Chen system is a special case of the Lorenz system

Abstract

In this paper, we show, by means of a linear scaling in time and coordinates, that the Chen system, given by ẋ=a(y-x), ẏ=(c-a)x+cy-xz, ż=-bz+xy, is, generically (c≠0), a special case of the Lorenz system. First, we infer that it is enough to consider two parameters to study its dynamics. Furthermore, we prove that there exists a homothetic transformation between the Chen and the Lorenz systems and, accordingly, all the dynamical behavior exhibited by the Chen system is present in the Lorenz system (since the former is a special case of the second). We illustrate our results relating Hopf bifurcations, periodic orbits, invariant surfaces, and chaotic attractors of both systems. Since there has been a large literature that has ignored this equivalence, the aim of this paper is to review and clarify this field. Unfortunately, a lot of the previous papers on the Chen system are unnecessary or incorrect. © 2013 AIP Publishing LLC.