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

Zigen Song,Weiguo Qian,Bin Zhen,Xianghong Kong

doi:10.1186/s13662-019-2066-8

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.

Highlights

Hopfield neural network was firstly proposed by Hopfield in 1984 [1]
We illustrated a multi-equilibria coexistence in a delayed Hopfield two-neural system with monotonic activation function
The results have shown that the neural system has two pitchfork bifurcations of the trivial equilibrium

Summary

Introduction

Hopfield neural network was firstly proposed by Hopfield in 1984 [1]. From on, the neural network systems have seen great development, both regarding their properties and applications, such as in pattern recognition, signal processing, and associative memory [2, 3]. For the low-dimensional Hopfield neural network system, the existing dynamical analysis is focused on the local stability and Hopf bifurcation of the trivial equilibrium [17,18,19]. The low-dimensional Hopfield neural system may exhibit the multi-coexistence of equilibria and periodic orbits [28, 29]. Song et al [30] employed the multistage pitchfork bifurcations of trivial and nontrivial equilibrium to find the multiple coexistences of stable and unstable equilibria in the Wilson–Cowan coupled system, which is a global dynamical analysis. The Hopfield neural network system exhibits two pitchfork bifurcations near the trivial equilibrium. 4, we will exhibit the stability analysis of the trivial equilibrium and find the periodic orbit using the Hopf bifurcation.

Pitchfork bifurcation of the trivial equilibrium

Conclusion