Zero-Hopf Bifurcation of a memristive synaptic Hopfield neural network with time delay

Abstract

This paper proposes a novel memristive synaptic Hopfield neural network (MHNN) with time delay by using a memristor synapse to simulate the electromagnetic induced current caused by the membrane potential difference between two adjacent neurons. First, some sufficient conditions of zero bifurcation and zero-Hopf bifurcation are obtained by choosing time delay and coupling strength of memristor as bifurcation parameters. Then, the third-order normal form of zero-Hopf bifurcation is obtained. By analyzing the obtained normal form, six dynamic regions are found on the plane with coupling strength of memristor and time delay as abscissa and ordinate. There are some interesting dynamics in these areas, i.e., the coupling strength of memristor can affect the number and dynamics of system equilibrium, time delay can contribute to both trivial equilibrium and non-trivial equilibrium losing stability and generating periodic solutions.

Keywords: Memristive Hopfield neural network (MHNN); Memristive synapse; Periodic solutions; Stability; Zero-Hopf bifurcation.