On the dynamics of the singularly perturbed of the difference equation with continuous arguments corresponding to the Hénon map

Abstract

The Hénon map was introduced by Hénon. It is a very rich dynamical model as Hénon himself proved and compared the dynamical properties of his map with those of other dynamical systems such as the Ro¨ssler model. Here, we study the dynamics of the difference equation with continuous arguments corresponding to the Hénon map and its singularly perturbed counterpart. The local stability of fixed points is studied. The system exhibits various types of bifurcation, such as saddle-node and Hopf bifurcations. By letting the perturbation parameter ∊⟶0, we show that the singularly perturbed equation exhibits the same qualitative behavior as its corresponding difference equation. The singularly perturbed equation exhibits the same qualitative behavior as its corresponding delay differential equation when ∊⟶1. The method of steps is used to discretize the original system to simulate the behavior of the original system. Numerical simulations are performed to confirm the theoretical analysis obtained and to illustrate the complex dynamics of the system. Moreover, the numerical simulations illustrate the effect of the perturbation parameter. In particular, the occurrence of halving-period bifurcation due to small change in the perturbation parameter.

Keywords

• Hénon map
• Singular perturbation
• stability
• Hopf bifurcation
• Saddle-node bifurcation
• Chaos