New Results to a Three Dimensional Chaotic System With Two Different Kinds of Nonisolated Equilibria

Abstract

In the paper by Liu et al. (2009, “A Novel ThreeDimensional Autonomous Chaos System,â€‌ Chaos Solitons Fractals, 39(4), pp. 1950–1958), the threedimensional (3D) chaotic system xآ·=axey2,yآ·=bykxz,zآ·=cz+mxy is investigated, and some of its dynamics according to theoretical and numerical analyses only for the parameters (a, e, b, k, c, m) = (1, 1, 2.5, 4, 5, 4) are discussed. In 2013, the same chaotic system xآ·1=ax1 fx2x3,xآ·2=cx2dx1x3,xآ·3=bx3+ex22 by Li et al. (2013, “Analysis of a Novel ThreeDimensional Chaotic System,â€‌ Optik, 124(13), pp. 1516–1522) was mainly discussed by numerical simulation. In this article, by some deeper investigations, combining some numerical simulations, we formulate some new results of the system. First, after some problems in the first paper are pointed out, we display that its parameters e, k, and m may be kicked out by some homothetic transformations. Second, some of its rich nonlinear dynamics hiding and not found previously, such as the stability and Hopf bifurcation of its isolated equilibria, the behavior of its nonisolated equilibria, the existence of singular orbits (including singularly degenerate heteroclinic cycle, homoclinic and heteroclinic orbits, etc.), and its dynamics at infinity, etc., are clearly formulated. What's more interesting, we find, this system has two different kinds of nonisolated equilibria Ex and Ez, and new chaotic attractors can be bifurcated out with the disappearance of Ex, but this system has no such properties at Ez. In the meantime, several problems about the existence of singular orbits deserving further investigations are presented. Our results better complement and improve the known ones.