Novel method to detect Hopf bifurcation in a delayed fractional-order network model with bidirectional ring structure

In this paper,we formulate and study a fractional-order network model with four neurons,bidirectional ring structure and self-delay feedback.For the scenario of nonidentical neurons,we develop a new algebraic technique to deal with the characteristic equation with e-4st(T is the self-feedback delay)term and thus establish the easy-tocheck criteria to determine the Hopf bifurcation point of self-feedback delay by fixing communication delay in its stable interval.For the scenario of identical neurons,we apply the crossing curves method to the fractional functional equations and thus procure the Hopf bifurcation curve.The obtained results accommodate the fact that the model cannot preserve its stability behavior when the self-feedback delay crosses the Hopf bifurcation point in the positive direction.Finally,we deliberate on the correctness of our methodology through two demonstration examples.

Stability and bifurcation analysis of a fractional-order single-gene regulatory model with delays under a novel PD^α control law

In this paper,we propose a novel fractional-order proportional-derivative(PD)strategy to achieve the control of bifurcation of a fractional-order gene regulatory model with delays.The stability theory of fractional differential equations proved that with delays,some explicit conditions for the local asymptotical stability and Hopf bifurcation are given for the controlled fractional-order genetic model.It is demonstrated that the fractional-order gene regulatory model becomes controllable by adjusting the control gain parameters.In addition,the effect of fractional-order parameter on the dynamical behaviors is shown.Finally,numerical simulations are carried out to testify the validity of the main results and the availability of the fractional-order PD controller.