Multiperiodicity Evoked by Periodic External Inputs in Cohen-Grossberg-type BAM Networks with Discrete and Distributed Delays

Abstract

In this paper, by using contraction mapping theorem, analysis approach and decomposition of solution space, the multiperiodicity issue is discussed for Cohen-Grossberg-type (CG-type) bidirectional associative memory networks (BAMNs) with discrete and distributed delays and a general class of activation functions, where the general class of activation functions consist of nondecreasing functions with saturations including piecewise linear functions with two corner points and standard activation functions as their special cases. It is shown that for any saturation region, if there is a periodic orbit located in it, it must be locally exponentially stable. Then, based on this result, some conditions are derived for ascertaining the (n + m)-neuron CG-type BAMNs can have 2 locally exponentially stable limit cycles located in two saturation regions respectively which are symmetrical. Also, taking account of different saturation regions, results about 2(p + q) (p ≤ m, q ≤ n − 1), 2min{n,m} locally exponentially stable limit cycles can be obtained, where n is the number of the neurons in one layer, m is the number of the neurons in the other layer. Meanwhile, for every locally exponentially stable limit cycle given, the corresponding saturation region can be expressed concretely. Finally, three examples are given to illustrate the effectiveness of the obtained results.