Dynamic analysis of Hopfield neural network with Sigmoid-type activation function

Abstract The Sigmoid function is a continuous and smooth nonlinear function, often called a logic function that exhibits an ‘S’-shaped characteristic similar to the Tanh function, yet possesses a distinct positive threshold property that differentiates it from the Tanh function. To elaborate on the special dynamic behaviors induced by such functions, this paper proposes a Sigmoid-type Hopfield neural network (Sigmoid-HNN) using a Sigmoid function as the activation function. With the Sigmoid-HNN model, the boundedness is proven using the Lyapunov method, and the stability of the equilibrium point is clarified exhaustively. The complex kinetic behaviors influenced by the synaptic weights and external current stimulation are analyzed using two-parameter bifurcation diagrams, phase diagrams, bifurcation diagrams, and Lyapunov exponential spectra. The results show that the bifurcation behaviors of the Sigmoid-HNN model exhibit remarkable similarity when adjusting the weight parameters and external stimuli; chaos crises frequently occur, and backward period-doubling bifurcation phenomena occur in both cases. In addition, PSIM was used to simulate an analog circuit to verify the response characteristics of the model. The simulation results are consistent with the experimental data, strongly confirming the feasibility of the system model at the physical level.