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

In the paper by Liu et al. (2009, “A Novel Three-Dimensional Autonomous Chaos System,” Chaos Solitons Fractals, 39(4), pp. 1950–1958), the three-dimensional (3D) chaotic system $x·=-ax-ey2,y·=by-kxz,z·=-cz+mxy$ 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 $x·1=-ax1- fx2x3,x·2=cx2-dx1x3,x·3=-bx3+ex22$ by Li et al. (2013, “Analysis of a Novel Three-Dimensional Chaotic System,” 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 E_x and E_z, and new chaotic attractors can be bifurcated out with the disappearance of E_x, but this system has no such properties at E_z. 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.