Singular Orbits and Dynamics at Infinity of a Conjugate Lorenz-Like System

Abstract

A conjugate Lorenz-like system which includes only two quadratic non-linearities is proposed in this paper. Some basic properties of this system, such as the distribution of its equilibria and their stabilities, the Lyapunov exponents, the bifurcations are investigated by some numerical and theoretical analysis. The forming mechanisms of compound structures of its new chaotic attractors obtained by merging together two simple attractors after performing one mirror operation are also presented. Furthermore, some of its other complex dynamical behaviours, which include the existence of singularly degenerate heteroclinic cycles, the existence of homoclinic and heteroclinic orbits and the dynamics at infinity, etc, are formulated in detail. In the meantime, some problems deserving further investigations are presented.

Keywords:

• Hopf bifurcation
• singularly degenerate heteroclinic cycle
• homoclinic and heteroclinic orbits
• Poincaré compactification

AMS Subject Classification:

• 34C23
• 34C37
• 34D08
• 34D20