Absorbing Basins and Relaxation Oscillation of Hopfield Neural Networks with Slow-fast Variables

On the basis of geometric singular perturbation theory, the dynamical behaviors of Hopfield neural networks with slow-fast variables are investigated.Firstly, the structure of the slow manifold is determined through the stability analysis of the fast subsystem.Then, the solution trajectory in the vicinity of the slow manifold is analysed by using the method of geometric singular perturbation, and the existence of relaxation oscillation is proved.Moreover, the period of the relaxation oscillation is calculated.When there are some stable equilibrium points in the slow manifold, the relaxation oscillation vanishes, and the shape of absorbing basin of every stable equilibrium points is determined.At last, numerical examples are given to validate analytical results.