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

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: x ˙ = a ( y - x ) , y ˙ = b 1 y + b 2 y z + b 3 x z + b 4 x 2 y , z ˙ = - c z + 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 , x 2 c ) | x ∈ R , c ≠ 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.