A hybrid control strategy incorporating auxiliary variables for active Hopf bifurcation relocation in fractional-order dynamical systems

To relocate the onset of system Hopf bifurcation to any target position, this paper proposes a novel hybrid controller based on state feedback. Firstly, by introducing auxiliary variables and incorporating fractional-order derivatives, a new fractional-order control strategy is formulated. Unlike traditional controllers that can only passively obtain bifurcation thresholds, the controller presented in this paper can preemptively identify the system's bifurcation points, thereby deriving relevant gain parameters, demonstrating greater proactivity and practical value. Subsequently, based on the fractional-order stability and Hopf bifurcation theory, the explicit expression for the precise value of the control gains corresponding to the target bifurcation location is successfully obtained. Finally, the effectiveness of this controller is verified numerically through two examples, during which it is discovered that not all control gains influence the onset of Hopf bifurcation. Meanwhile, the relationship between the expected bifurcation points and the control gains is investigated.