Hopf bifurcation analysis in a fractional-order survival red blood cells model and PD α $\mathit{PD}^{\alpha} $ control

Abstract

Abstract In this paper, we put forward a fractional-order survival red blood cells model and study the dynamics through the Hopf bifurcation. When the delay transcends the threshold, a series of Hopf bifurcations occur at the positive equilibrium. Then, a fractional-order Proportional and Derivative ( PD α $\mathit{PD}^{\alpha} $ ) controller is applied to the proposed model for the Hopf bifurcation control. It is discovered that by setting proper parameters, the PD α $\mathit{PD}^{\alpha} $ controller can delay or advance the onset of Hopf bifurcations. Therefore the Hopf bifurcation of the fractional-order survival red blood cells model becomes controllable to achieve desirable behaviors. Finally, numerical examples are presented to demonstrate the theoretical analysis.

Keywords

• Hopf bifurcation
• bifurcation control
• time delays
• PD α $\mathit{PD}^{\alpha} $ controller
• survival red blood cells model