Matrix criterion for dynamic analysis in discrete neural networks with multiple delays

The dynamics of a discrete Hopfield neural network with multiple delays (HNNMDs) is studied by using a matrix inequality which is shown to be equivalent to the state transition equation of the HNNMDs network. Earlier work (2000) on discrete Hopfield neural networks showed that a parallel or serial mode of operation always leads to a limit cycle of period one or two for a skew or symmetric matrix, but they did not give an arbitrary weight matrix on how an updating operation might be needed to reach such a cycle. In this paper we present the existence conditions of limit cycles using matrix criteria in the HNNMDs network. For a network with an arbitrary weight matrix, the necessary and sufficient conditions for the existence of a limit cycle of period 1 and r are provided. The conditions for the existence of a special limit cycle of period 1 and 2 are also found. These results provide the foundation for many applications. A HNNMDs is said to have no stable state (fixed point) if it has a limit cycle of period 2 or more, which is stated in Theorem 5. A computer simulation demonstrates that the theoretical analysis in Theorem 5 is correct.