Stability switches and Bogdanov-Takens bifurcation in an inertial two-neuron coupling system with multiple delays

In this paper, we investigate an inertial two-neural coupling system with multiple delays. We analyze the number of equilibrium points and demonstrate the corresponding pitchfork bifurcation. Results show that the system has a unique equilibrium as well as three equilibria for different values of coupling weights. The local asymptotic stability of the equilibrium point is studied using the corresponding characteristic equation. We find that multiple delays can induce the system to exhibit stable switching between the resting state and periodic motion. Stability regions with delay-dependence are exhibited in the parameter plane of the time delays employing the Hopf bifurcation curves. To obtain the global perspective of the system dynamics, stability and periodic activity involving multiple equilibria are investigated by analyzing the intersection points of the pitchfork and Hopf bifurcation curves, called the Bogdanov-Takens (BT) bifurcation. The homoclinic bifurcation and the fold bifurcation of limit cycle are obtained using the BT theoretical results of the third-order normal form. Finally, numerical simulations are provided to support the theoretical analyses.