Bifurcations and chaos in a three-dimensional generalized Hénon map

Abstract

Abstract This article presents the bifurcation and chaos phenomenon of the three-dimensional generalized Hénon map. We establish the existence and stability conditions for the fixed points of the system. According to the center manifold theorem and bifurcation theory, we get the existence conditions for fold bifurcation, flip bifurcation, and Naimark–Sacker bifurcation of the system. Finally, the bifurcation diagrams, Lyapunov exponents, phase portraits are carried out to illustrate these theoretical results. Furthermore, as parameter varies, new interesting dynamics behaviors, including from stable fixed point to attracting invariant cycle and to chaos, from periodic-10 to chaos, etc., are observed from the numerical simulations. In particular, we find the double-cycle phenomenon from bifurcation diagrams and phase portraits.

Keywords

• Hénon map
• Fold bifurcation
• Flip bifurcation
• Naimark–Sacker bifurcation
• Double-cycle