Multiple bifurcations and periodic coexistence in a delayed Hopfield two-neural system with a monotonic activation function

Abstract

Abstract In this paper, we consider a delayed Hopfield two-neural system with a monotonic activation function and find the periodic coexistence by bifurcation analysis. Firstly, we obtain the pitchfork bifurcation of the trivial equilibrium employing the central manifold and normal form methods. The neural system exhibits two pitchfork bifurcations near the trivial equilibrium. Then, analyzing the characteristic equation of the nontrivial equilibrium, we illustrate the saddle-node bifurcation of the nontrivial equilibria. The system exhibits the multi-coexistences of the stable and unstable equilibria. Further, we illustrate the plane regions of parameters having different numbers of equilibria. To obtain a time delay in neural system dynamics, we present the stability analysis and find the periodic orbit. The system exhibits stability switching by the Hopf bifurcation curves. Finally, the dynamic behaviors near the Hopf–Hopf bifurcation point are presented. The system exhibits coexistence of multiple periodic orbits with different frequencies.

Keywords

• Hopfield neural system
• Multiple delays
• Coexistence
• Multiple bifurcations
• Monotonic activation function