Department of Mathematics, Facultad de Ciencias Experimentales, University of Huelva, Avda. Tres de Marzo s/n, 21071 Huelva, Spain.
Chaos. 2013 Sep;23(3):033108. doi: 10.1063/1.4813227.
In this paper, we show, by means of a linear scaling in time and coordinates, that the Chen system, given by x=a(y-x), y=(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.
在本文中，我们通过时间和坐标的线性标度证明，Chen 系统，其方程为 x=a(y-x)，y=(c-a)x+cy-xz，ż=-bz+xy，一般情况下（c≠0），是 Lorenz 系统的一个特例。首先，我们推断出研究其动力学只需要考虑两个参数。此外，我们证明了 Chen 系统和 Lorenz 系统之间存在同胚变换，因此，Chen 系统所表现出的所有动力学行为都存在于 Lorenz 系统中（因为前者是后者的特例）。我们说明了与两个系统的 Hopf 分岔、周期轨道、不变曲面和混沌吸引子相关的结果。由于大量文献忽略了这种等价性，本文的目的是回顾和澄清这一领域。不幸的是，之前关于 Chen 系统的许多论文都是不必要的或不正确的。
