Post-Double Hopf Bifurcation Dynamics and Adaptive Synchronization of a Hyperchaotic System

Abstract

In this paper a four-dimensional hyperchaotic system with only one equilibrium is\ud considered and its double Hopf bifurcations are investigated. The general post-bifurcation\ud and stability analysis are carried out using the normal form of the system obtained via\ud the method of multiple scales. The dynamics of the orbits predicted through the normal\ud form comprises possible regimes of periodic solutions, two-period tori, and three-period\ud tori in parameter space.\ud Moreover, we show how the hyperchaotic synchronization of this system can be realized\ud via an adaptive control scheme. Numerical simulations are included to show the\ud effectiveness of the designed control

The chaotic Dadras–Momeni system: control and hyperchaotification

“…The intrinsic nature of the chaotic systems does not obey to this idea of synchronization due to the sensitivity to initial conditions; in fact the trajectories of two identical chaotic systems evolve to completely different dynamical behavior when starting from different initial conditions (however they are close). However, the interest in synchronization schemes of chaotic systems is also rapidly increasing [1,14,27,34] due to important application in secure communications [2,38,3] and cryptography [35]. Here we design a nonlinear controller according to Lyapunov's direct method to guarantee a complete and global synchronization between two identical Dadras-Momeni systems in their chaotic regime.…”