Analytical study of global bifurcations, stabilization and chaos synchronization of jerk system with multiple attractors

Abstract

The aim of this paper is to extend the recent analytical study of local bifurcations of a chaotic jerk model with multiple attractors to global bifurcations and examine the chaos synchronization problem for the case of multiple attractors and unknown system’s parameters. In particular, the different types of bifurcations of limit cycles exist in the model are explored analytically. The range of values in three-dimensional space of parameters, corresponding to each type of bifurcation, is found. A combination of time domain and frequency domain techniques, including multiple scales perturbation method and harmonic balance method, is employed in order to achieve this goal. More specifically, the study reveals that applying both multiple scales method and describing function method in a hybrid scheme enhances the accuracy of estimated critical bifurcation values as well as overcomes the problem of multiple scales method in capturing the correct values for bifurcation. Moreover, the chaos synchronization can be achieved in spite of the existence of multiple attractors. Finally, stabilization of fixed points and some periodic orbits of the system are studied using time-delayed feedback control scheme. Numerical simulations are presented so as to verify theoretical results.