New route of chaotic behavior in a 3D chaotic system

Abstract

This article revisits a three-dimensional Lorenz-like system x˙=a(y−x), y˙=bx−lxz, z˙=−cz+hx2+ky2 presented in Liu et al. (2006), where only the parameter values (a, b, l, c, h, k)=(10, 40, 1, 2.5, 2, 2) and the initial value (x0, y0, z0)=(2.2, 2.4, 28) are considered. One here not only finds that this system possesses new chaotic route: from stability directly to chaos, but also mathematically obtains some of its other wonderful dynamics, for example, its local dynamics including the stability and Hopf bifurcation of its isolated equilibria and the behavior of its non-isolated equilibria, its global dynamics including singularly degenerate heteroclinic cycle, homoclinic and heteroclinic orbits, and its dynamics at infinity, etc. Numerical simulations also display the new route of chaotic behavior.