Periodic and homoclinic solutions of discontinuous Cohen–Grossberg neural networks with time-varying delays

Abstract

In this paper, we investigate a class of discontinuous Cohen–Grossberg neural networks (DCGNNs) with time-varying delays. Under the framework of Filippov solution, by means of differential inclusions theory and the set-valued version of the Mawhin coincidence theorem, we firstly establish the existence results of 2kT-periodic solutions for the proposed DCGNNs. Secondly, by applying an approximation technique, we prove that the limit of the 2kT-periodic solutions exists and the limit is exactly the homoclinic solution of the proposed DCGNNs. Thirdly, by using the non-smooth analysis theory with Lyapunov-like approach, some new testable algebraic criteria are derived for ensuring the global exponential stability of the solutions. It should be regarded as the first time to study the homoclinic solutions of Cohen–Grossberg neural networks with discontinuous activations based on Filippov solution. Finally, convincing numerical examples and remarks are provided to substantiate the superiority and efficiency of obtained results.