Codimension-two bifurcation and multistability coexistence in an inertial two-neuron system with multiple delays

Employing the codimension-two bifurcation analysis, multistability coexistence and the corresponding parameter regions can be exhibited in the nonlinear dynamical systems. In this paper, an inertial two-neuron system with multiple delays is established and the dynamic behaviors are studied. The neural system exhibits one/three equilibria by the pitchfork bifurcation. Analyzing the Hopf bifurcation, we found that multiple delays can induce system dynamics to exhibit the stability switching. Further, the stability regions with the delayed dependent/independent are illustrated, followed which one of the codimension-two bifurcations, i.e., the Hopf–Hopf bifurcation is obtained from the intersection points of the Hopf bifurcation curves. The dynamic behavior near the bifurcation singularity is analyzed. It follows that the original system exhibits the stability coexistence with two different periodic activities. Finally, the other codimension-two bifurcation, i.e., the Pitchfork–Hopf bifurcation is investigated, where the system characteristic equation has a zero and a pair of pure imaginary eigenvalues. Theoretical results and numerical simulations are illustrated to show neural system exhibits the stability coexistence with a pair of nontrivial equilibrium and a periodic activity.

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