A class of simple Hopfield neural networks with a parameter is investigated. Numerical simulations show that the simple Hopfield neural networks can display chaotic attractors and limit cycles for different parameters. The Lyapunov exponents are calculated; the bifurcation plot and several important phase portraits are presented as well. By virtue of a recent result of horseshoe theory in dynamical systems, we present rigorous computer-assisted verifications for chaotic behavior in the simple Hopfield neural networks for certain parameters and give a brief discussion on the robustness of the chaotic behavior. Quantitative descriptions of the complexity of these neural networks are also given in terms of topological entropy.

This study is concerned with the bounded synchronisation of singularly perturbed complex network. If the reduced network is boundedly synchronised, the authors obtain, by using partially contracting theory, an explicit bound for small perturbation parameter to guarantee the bounded synchronisation of singularly perturbed complex network. Moreover, the authors present a sufficient condition such that the reduced network is boundedly synchronised. Although this sufficient condition is difficult to test directly for general networks, it can be tested for power networks based on some existing results. The effectiveness of the proposed results is demonstrated via power networks.