On Hopf bifurcation in fractional dynamical systems

Fractional order dynamical systems admit chaotic solutions and the chaos disappears when the fractional order is reduced below a threshold value [1]. Thus the order of the dynamical system acts as a chaos controlling parameter. Hence it is important to study the fractional order dynamical systems and chaos. Study of fractional order dynamical systems is still in its infancy and many aspects are yet to be explored. In pursuance to this in the present paper we prove the existence of fractional Hopf bifurcation in case of fractional version of a chaotic system introduced by Bhalekar and Daftardar-Gejji [2]. We numerically explore the (A, B, α) parameter space and identify the regions in which the system is chaotic. Further we find (global) threshold value of fractional order α below which the chaos in the system disappears regardless of values of system parameters A and B.