Modeling, dynamical analysis and numerical simulation of a new 3D cubic Lorenz-like system

Abstract

Little seems to be considered about the globally exponentially asymptotical stability of parabolic type equilibria and the existence of heteroclinic orbits in the Lorenz-like system with high-order nonlinear terms. To achieve this target, by adding the nonlinear terms yz and x^{2}y to the second equation of the system, this paper introduces the new 3D cubic Lorenz-like system: dot{x}=a(y - x), dot{y}=b_{1}y+b_{2}yz+b_{3}xz+b_{4}x^{2}y, dot{z}= -cz + y^{2}, which does not belong to the generalized Lorenz systems family. In addition to giving rise to generic and degenerate pitchfork bifurcation, Hopf bifurcation, hidden Lorenz-like attractors, singularly degenerate heteroclinic cycles with nearby chaotic attractors, etc., one still rigorously proves that not only the parabolic type equilibria S_{x} = {(x, x, frac{x^{2}}{c})|xin mathbb {R}, cne 0} are globally exponentially asymptotically stable, but also there exists a pair of symmetrical heteroclinic orbits with respect to the z-axis, as most other Lorenz-like systems. This study may offer new insights into revealing some other novel dynamic characteristics of the Lorenz-like system family.