Hopf-pitchfork bifurcation in a two-neuron system with discrete and distributed delays

Both discrete and distributed delays are considered in a two-neuron system. We analyze the influence of interaction coefficient and time delay on the Hopf-pitchfork bifurcation. First, we obtain the codimension-2 unfolding with original parameters for Hopf-pitchfork bifurcation by using the center manifold reduction and the normal form method. Next, through analyzing the unfolding structure, we give complete bifurcation diagrams and phase portraits, in which multistability and other dynamical behaviors of the original system are found, such as a stable periodic orbit, the coexistence of two stable nontrivial equilibria, and the coexistence of a stable periodic orbit and two stable equilibria. In addition, the obtained theoretical results are verified by numerical simulations. Finally, we perform the comparisons of the obtained results of Hopf-pitchfork bifurcation with other Hopf-fold bifurcation results in some biological neural systems and give the obtained mathematical results corresponding to the physical states of neurons.