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

Abstract

AbstractLittle 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 termsyzand$$x^{2}y$$x2yto the second equation of the system, this paper introduces the new 3D cubic Lorenz-like system:$$\dot{x}=a(y - x)$$x˙=a(y-x),$$\dot{y}=b_{1}y+b_{2}yz+b_{3}xz+b_{4}x^{2}y$$y˙=b1y+b2yz+b3xz+b4x2y,$$\dot{z}= -cz + y^{2}$$z˙=-cz+y2, 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$$S_{x} = \{(x, x, \frac{x^{2}}{c})|x\in \mathbb {R}, c\ne 0\}$$Sx={(x,x,x2c)|x∈R,c≠0}are globally exponentially asymptotically stable, but also there exists a pair of symmetrical heteroclinic orbits with respect to thez-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.