Cluster synchronization of coupled delayed competitive neural networks with two time scales

This paper investigates the cluster synchronization problem of coupled delayed competitive neural networks (CNNs) with two time scales. Each CNN contains short- and long-term memories, which can be regarded as the fast and slow dynamics, respectively. Besides, a general communication topology that describes both cooperation and competition in CNN-to-CNN relations is considered along with fixed and adaptive coupling schemes. The interactive relationship between the fast and slow dynamics as well as the effects of the fast time scale on synchronization behavior has not been fully exploited in existing Lyapunov functionals. Moreover, the results from pervious works are limited to the master–slave synchronization of two CNNs. In this paper, a novel Lyapunov–Krasovskii functional is proposed to solve the cluster synchronization problem under the fixed coupling scheme. The coupled delayed CNNs within a specific range of the fast time scale achieve a desirable behavior when the coupling and pinning strengths are chosen properly. Furthermore, to facilitate the selection of these strengths, an adaptive pinning controller is designed and a modified Lyapunov–Krasovskii functional is also constructed for coupled delayed CNNs with two time scales. Finally, several numerical examples are provided to demonstrate the effectiveness of the theoretical results.