Hopf–pitchfork bifurcation in an inertial two-neuron system with time delay

In this paper, we have considered a delayed differential equation modeling two-neuron system with both inertial terms and time delay. By studying the distribution of the eigenvalues of the corresponding transcendental characteristic equation of the linearization of this equation, we derive the critical values where Hopf–pitchfork bifurcation occurs. Then, by computing the normal forms for the system, the bifurcation diagrams are obtained. Furthermore, we find some interesting phenomena, such as the coexistence of two asymptotically stable states, two stable periodic orbits, and two attractive quasi-periodic motions, which are verified both theoretically and numerically.