Multistability and multiperiodicity of high-order competitive neural networks with a general class of activation functions

In this paper, high-order synaptic connectivity is introduced into competitive neural networks and the multistability and multiperiodicity issues are discussed for high-order competitive neural networks with a general class of activation functions. Based on decomposition of state space, Halanay inequality, Cauchy convergence principle and inequality technique, some sufficient conditions are derived for ascertaining equilibrium points to be located in any designated region and to be locally exponentially stable. As an extension of multistability, some similar results are presented for ensuring multiple periodic solutions when external inputs and time delay are periodic. The obtained results are different from and less restrictive than those given by Nie and Cao (2009 [25]), and the assumption (H1A) by Nie and Cao (2009 [25]) is relaxed. It is shown that high-order synaptic connectivity plays an important role on the number of equilibrium points and their dynamics. As a consequence, our results refute traditional viewpoint: high-order synaptic connectivity has faster convergence rate and greater storage capacity than first-order one. Finally, three examples with their simulations are given to show the effectiveness of the obtained results.