Existence and Exponential Stability of Multiple Periodic Solutions for a Multidirectional Associative Memory Neural Network

Abstract

The paper proposes several mathematical models of the multidirectional associative memory (MAM) neural network by analyzing its structure. A model of MAM with distributed delays is studied. Under some new assumptions on activation functions, 2 n 0 [m/2] invariant subsets of MAM are constructed. Then the existence and the exponential stability of 2 n 0 [m/2] periodic solutions located on invariant subsets are obtained by constructing a suitable Liapunov function and a Poincaré mapping. An estimating method of the exponential convergence rate is given. The obtained results are new to MAM neural networks. An example is given to illustrate the effectiveness of the results.

Multiple periodic solutions of delayed competitive neural networks via functional differential inclusions

“…Remark 2. In the earlier papers, a series of results on the multiperiodicity for neural networks with continuous neuron activation functions were obtained [10,[17][18][19]. However, for neural networks with discontinuous neuron activation functions, there are few papers consider the multiperiodicity of them.…”

“…Therefore, the methods used in the papers [10,[17][18][19] cannot be applied to discuss the existence of multiple periodic solutions to the system (4.2). To deal with the existence of multiple periodic solutions (or equilibrium) of neural networks systems with general discontinuous neuron activations functions, the methods in this paper are very effective and thus the results of this paper (Theorems 3.3 and 3.4) are essentially new.…”

“…For a class of activation functions consist of nondecreasing functions with saturation's including piecewise linear functions with two corner points and standard activation functions as its special case, Cao et al discussed the multiperiodicity of delayed neural networks [18]. In [19], under the assumption that the activation functions are increasing and bounded, Zhou et al studied the multiperiodicity of multidirectional associative memory neural networks via Poincaré mapping.…”

Multistability and Multiperiodicity for a General Class of Delayed Cohen-Grossberg Neural Networks with Discontinuous Activation Functions

“…So it is necessary that there exist multiple stable equilibria or periodic orbits for neural networks, which are usually referred to as multistability or multiperiodicity, respectively. In the last few years, the multistability and multiperiodicity of neural networks have been reported in [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] and the references therein. In particular, [14][15][16][17][18][19][20][21] investigated the multistability or multiperiodicity of neural networks with sigmoidal activation functions or nondecreasing saturated activation functions.…”

Multistability in a Multidirectional Associative Memory Neural Network with Delays

“…The stability of Hopfield neural networks and BAM neural networks is discussed in a lot of recently published literature works [8][9][10][11][12], but the researchers about MAM neural networks are mainly focused on learning algorithms, fault tolerance, and retrieval efficiency of MAM neural networks [3][4][5][6]. To the best of our knowledge, the research on the theory of MAM neural networks was reported only in a few papers [13][14][15][16][17][18]. Chen et al proved the stability of some specific types of MAM neural networks in [13,14].…”