Attractive Periodic Orbits in Nonlinear Discrete-time Neural Networks with Delayed Feedback |
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Authors: | Zhan Zhou Jianhong Wu |
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Institution: | 1. Department of Applied Mathematics , Hunan University , Changsha, Hunan, 410082, Peoples Republic of China;2. Department of Mathematics and Statistics , York University , Toronto, Ont., M3J 1P3, Canada |
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Abstract: | We consider the discrete-time system x ( n )= g x ( n m 1)+ f ( y ( n m k )), y ( n )= g y ( n m 1)+ f ( x ( n m k )), n ] N describing the dynamic interaction of two identical neurons, where g ] (0,1) is the internal decay rate, f is the signal transmission function and k is the signal transmission delay. We construct explicitly an attractive 2 k -periodic orbit in the case where f is a step function (McCulloch-Pitts Model). For the general nonlinear signal transmission functions, we use a perturbation argument and sharp estimates and apply the contractive map principle to obtain the existence and attractivity of a 2 k -periodic orbit. This is contrast to the continuous case (a delay differential system) where no stable periodic orbit can occur due to the monotonicity of the associated semiflow. |
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Keywords: | Attractivity Periodic Solution Discrete-time Neural Networks Delay |
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