Control of Julia Sets in Complex Henon Systems

Abstract

In 1976, the famous French astronomer, Michel Henon [433], proposed the famous two-dimensional iterative map,

$$\begin{aligned} \left\{ \begin{array}{l} x_{k+1}=1+y_k-ax_k^2,~a=1.4, \\ y_{k+1}=bx_k,~~~~~~~~~~~~~~~b=0.3. \end{array}\right. \end{aligned}$$

At present, there are many results regarding the Henon map, which broadly emerge in engineering applications, such as the basic comprehension of the notorious small denominator problem plaguing Hamiltonian mechanics and the scaling behavior of circle maps modeling successfully the onset of chaos in many real physical situations [434–444]. It is well known that the Julia set is an important notion in fractal theory and it has extensive applications in physics, biology, and so on. Especially, the strange attractor and the boundary of the attractive region of the recursive mapping on the complex plane C are associated with chaotic dynamical systems.