Dissipative feedback control and synchronization of Julia sets of the complex standard family

In this paper, control of Julia set of the complex standard family zn+1=zn+α+βsin(zn) is studied by adding the dissipative feedback control item to the system in two different ways. One is added to the complex system directly and the other is added to the imaginary part of the complex system. For this control method, both the system and the fixed point can be controlled at the same time even the unstable periodic orbits or the fixed points are unknown. Synchronization of two different Julia sets of this system with different system parameters is discussed by two methods, which makes one Julia set change to be another one.