Local bifurcation analysis and topological horseshoe of a 4D hyper-chaotic system

In this paper, a 4D autonomous system with complex hyper-chaotic dynamics is introduced. The Lyapunov exponent spectrum, bifurcation diagram and phase portrait are provided. Basic dynamical properties are also analyzed. In order to clarify the evolution of the complex dynamic behaviors of the system, the local bifurcation is studied and a Hopf bifurcation is proved to occur when the appropriate bifurcation parameter passes the critical value. All the conditions of Hopf bifurcation are derived by applying center manifold theorem and Poincar,-Andronov-Hopf bifurcation theorem. Numerical simulation results are consistent with the theoretical analysis. Besides, we present a rigorous study on the hyper-chaotic system by combining the topological horseshoe theory with a computer-assisted approach of Poincar, maps. Utilizing the algorithm for finding horseshoes in 3D hyper-chaotic maps, a horseshoe with two-directional expansion in the 4D hyper-chaotic system has been found, which rigorously proves the existence of hyper-chaos in theory.