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.