Conditions for the global asymptotic stability of delayed artificial neural network model of n (≥3) neurons have been derived. For bifurcation analysis with respect to delay we have considered the model with three neurons and used suitable transformation on multiple time delays to reduce it to a system with single delay. Bifurcation analysis is discussed with respect to single delay. Numerical simulations are presented to verify the analytical results. Using numerical simulation, the role of delay and neuronal gain parameter in changing the dynamics of the neural network model has been discussed. 1. Introduction In recent years, neural networks (especially Hopfield type, cellular, bidirectional, and recurrent neural networks) have been applied successfully in many areas such as signal processing, pattern recognition, and associative memories. In most of the research works, stability of the designed neural network is an important step of analyzing the dynamics. Chaos plays an important role in human brain cognitive functions related to memory process. For example, chaotic behavior has been observed in nerve membranes by electrophysiological experiments on squid giant axons [1–3] and in measurements of brain electroencephalograms (EEG) [4, 5]. At first, Aihara et al. [6] introduced chaotic neural network models in order to simulate the chaotic behavior of biological neurons. Both the network and its component neuron are responsible for chaotic dynamics if suitable parameter values are chosen [6, 7]. The investigation of chaotic neural networks is of practical importance and many interesting results have been obtained so far (see [8–11] and the references therein). The control of chaotic behavior in chaotic neural networks is an important problem to apply them in information processing [12]. The first chaos control was proposed by Ott et al. (the OGY method) [13]. Since this pioneer work of OGY, various methods such as the occasional proportional feedback (OPF) method [14], continuous feedback control [15], and pinning method [16] have been proposed for chaos control. Many research works have been done to know the effect of time delays in neural system [17–19]. The malfunctioning of the neural system is often related to changes in the delay parameter causing unmanageable shifts in the phases of the neural signals. In the olfactory system, the phase transition has the appearance of a change in the EEG from a chaotic, aperiodic fluctuation to a more regular nearly periodic oscillation. In fact, neural network with delay can actually synchronize more easily
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