Hopf bifurcation, chaos control and synchronization of a chaotic fractional-order system with chaos entanglement function

Abstract

This paper deals with the stability and bifurcation of equilibria in a new chaotic fractional-order system in the sense of the Caputo fractional derivative with the chaos entanglement function. We derive conditions under which the system undergoes a Hopf bifurcation and obtain critical parameter value in the Hopf bifurcation. Moreover, the linear feedback control technique is used to control and stabilize the system to equilibrium point in order to eliminate the chaotic vibration. We then design control laws to synchronize two identical chaotic fractional-order systems. Furthermore, by means of numerical simulation, we support the validity of analytical results and reveal more dynamical behaviors consisting chaos, local bifurcation, limit cycles, quasiperiodic and asymptotic stability behaviors. We further emphasize that the order of fractional derivative plays significant roles as the chaos controlling parameter and the Hopf bifurcation parameter.