Dynamics of the general Lorenz family

Abstract

The present work is devoted to investigating the dynamical entities of the general Lorenz family, which contains four independent parameters. The classical Lorenz system, the Chen system, and the Lu system are all contained by the system considered in this paper as special cases. First, the properties of the equilibria, in particular, the stability of the non-hyperbolic equilibrium obtained by using the center manifold theorem and the technique of the polar transformation, the pitchfork bifurcation and the degenerate pitchfork bifurcation, Hopf bifurcations, and the local stable and unstable manifold character, are all analyzed when the parameters are varied in the space of parameters. Based on the theoretic analysis and numerical simulations, the dynamics of the system are discussed subtly under all kind of the critical state. Second, the properties of the existence of homoclinic and heteroclinic orbits for the system are rigorously studied. Finally, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated.