Complex Dynamics in the Unified Lorenz-Type System

Abstract

This paper is devoted to the analysis of complex dynamics of the unified Lorenz-type system (ULTS) with six parameters, which contain common chaotic systems as its particular cases. First, some important local dynamics such as pitchfork bifurcation, Hopf bifurcation, and the stability of nondegenerate and double-zero equilibria are systematically investigated using the parameter-dependent center manifold theory combined with some bifurcation theories. Some adequate conditions for guaranteeing the occurrence of degenerate Hopf bifurcation (DHB) and the stability of the equilibria are given. Second, it is found that if DHB does not generate at the trivial equilibrium but generates at two symmetric nontrivial equilibria, then a small perturbation can lead that ULTS to exhibit a chaotic attractor. Interestingly, such a case can take place in the Chen and Lü systems (two common chaotic systems) but cannot take place in the Lorenz and Yang systems (the other two common chaotic systems), essentially distinguishing the Lorenz system from the Chen system. In addition, it is numerically verified that both of the latter two systems can exhibit the coexistence of both a chaotic attractor and multiple limit cycles but the former two systems seem not to have this property. If DHB takes place simultaneously at three equilibria of ULTS, then this system has an invariant algebraic surface, and rigorously prove the existence of some global dynamics such as periodic orbit, center, homoclinic/heteroclinic orbits. Third, it is shown that a singularly degenerate heteroclinic cycle can exist in the case of b = 0 (where b is a parameter of ULTS, like that in the Lorenz system), and a chaotic attractor can be generated by perturbing this cycle for small b > 0. These results altogether indicate that the ULTS can exhibit complex dynamics, and provide a more reasonable classification for chaos in the 3D autonomous chaotic ODE systems that were developed based on the Lorenz system, in contrast to the previous studies.