Coexisting behaviors of a fraction-order novel hyperbolic-type memristor Hopfield neuron network based on three neurons

Abstract

The activation function of human neurons is usually regarded as a monotonically differentiable function with upper and lower bounds. Considering the mathematical properties of the hyperbolic tangent function, the activation function can be simulated by a hyperbolic tangent function. In this paper, a fraction-order novel hyperbolic-type memristor Hopfield neuron network (FHMHNN) based on three neurons is proposed, which is achieved using a hyperbolic-type memristor synapse-coupled weight to substitute a coupling-connection weight. The equilibrium points and stability analysis of the FHMHNN are discussed in detail, and the types of generating attractor are determined. Furthermore, the coexisting behaviors of the FHMHNN are described by bifurcation diagram, phase diagram and time diagram. Numerical results show that the FHMHN presents complex dynamical transition, evolving from periodic to chaotic and finally to a stable point with the changes of the memristor coupling weight and inner parameter of the hyperbolic-type memristor. It should be emphasized that the coexisting oscillation behaviors of the FHMHNN under different initial conditions will appear for different inner parameters of the memristor. Theoretical analysis and numerical simulation are basically consistent, revealing that the FHMHNN has the globally coexisting behavior of the asymmetric attractors.