Abstract
Problem statement: In this study linear stability of a class of three neuron cellular network with transmission delay had been studied. Approach: The model for the problem was first presented. The problem is then formulated analytically and numerical simulations pertaining to the model are carried out. Results: A necessary and sufficient condition for asymptotic stability of trivial steady state in the absence of delay is derived. Then a delay dependent sufficient condition for local asymptotic stability of trivial, steady state and sufficient condition for no stability switching of trivial steady for such a network are derived. Numerical simulation results of the model were presented. Conclusion/Recommendations: From numerical simulation, it appears that there may be a possibility of multiple steady states of the model. It may be possible to investigate the condition for the existence of periodic solutions of the non-linear model analytically.
Highlights
The notion of Cellular Neural Network (CNNs) was introduced by Chua and Yang (1998) and since CNN models have been used in many engineering applications, e.g., in signal processing and especially in static image treatment (Chua, 1988)
Periodic solutions and exponential stability in delayed cellular network and sufficient conditions for global asymptotic stability of cellular neural network with delay are discussed by Cao (2000); Zhang et al (2007) and others (Gyori and Hartung, 2004) respectively
In this study we have considered a class of three- neuron delayed cellular neural network and have studied the local stability phenomenon of its trivial equilibrium
Summary
The notion of Cellular Neural Network (CNNs) was introduced by Chua and Yang (1998) and since CNN models have been used in many engineering applications, e.g., in signal processing and especially in static image treatment (Chua, 1988). The necessary and sufficient condition of local asymptotic stability of the trivial steady state (0,0,0) in absence of time delay has been derived. Length of time delay has been estimated, below which trivial steady state remains asymptotically stable.
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