Dynamics of a New Four-Thirds-Degree Sub-Quadratic Lorenz-like System

Abstract

Aiming to explore the subtle connection between the number of nonlinear terms in Lorenz-like systems and hidden attractors, this paper introduces a new simple sub-quadratic four-thirds-degree Lorenz-like system, where x˙=a(y−x), y˙=cx−x3z, z˙=−bz+x3y, and uncovers the following property of these systems: decreasing the powers of the nonlinear terms in a quadratic Lorenz-like system where x˙=a(y−x), y˙=cx−xz, z˙=−bz+xy, may narrow, or even eliminate the range of the parameter c for hidden attractors, but enlarge it for self-excited attractors. By combining numerical simulation, stability and bifurcation theory, most of the important dynamics of the Lorenz system family are revealed, including self-excited Lorenz-like attractors, Hopf bifurcation and generic pitchfork bifurcation at the origin, singularly degenerate heteroclinic cycles, degenerate pitchfork bifurcation at non-isolated equilibria, invariant algebraic surface, heteroclinic orbits and so on. The obtained results may verify the generalization of the second part of the celebrated Hilbert’s sixteenth problem to some degree, showing that the number and mutual disposition of attractors and repellers may depend on the degree of chaotic multidimensional dynamical systems.