Multi-parametric bifurcations of a fractional neural network with multiple delays and inertial terms

In this paper, the Hopf bifurcation of a Caputo fractional-order delayed neural network with inertial terms is systematically investigated. Time delays and fractional order are chosen as the bifurcation parameters, respectively, with an indirect discussion on how the parameters in the inertial terms affect stability. Firstly, the induction conditions for bifurcation caused by time delay are analyzed. The critical values of the characteristic equation are acquired by utilizing two nonidentical methods: degree reduction of transcendental terms and the implicit function array. Secondly, in compliance with the quadratic relationship of fractional order, the conditions for inducing bifurcation of fractional order and inertial parameter are extracted. In the last resort, two numerical simulations corroborate the theoretical results, and illustrate how time delay, fractional order, and the inertial parameter affect system stability. These parameters of reductions magnify the stability region and thereupon augment the stable operation.