Hopf-type bifurcation and synchronization of a fractional order Van der Pol oscillator

In this paper, the dynamical behaviors of an incommensurate fractional order Van der Pol oscillator are studied. First, based on the stability theory of fractional order systems, some sufficient conditions for the stability and Hpof-type bifurcation are given for such fractional order system. Then, the frequency and amplitude of periodic oscillations are determined by numerical simulations. It is shown that the frequency of oscillations gets smaller as the fractional order is increased, while the amplitude of oscillations increases with the fractional order. Finally, the synchronization of two coupled fractional order Van der Pol oscillators is analyzed with the help of the linear feedback control method. Numerical simulations are performed to verify the theoretical results.