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Complex dynamics of a four neuron network model having a pair of short-cut connections with multiple delays

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Abstract

This paper deals with dynamic behaviors on Hopfield type of ring neural network of four neurons having a pair of short-cut connections with multiple time delays. By suitable transformation and under certain assumptions on multiple time delays, the model is reduced to four dimensional nonlinear delay differential equations with three delays. Regarding these time delays as parameters, delay independent sufficient conditions for no stability switches of the trivial equilibrium of the linearized system are derived. Conditions for stability switching with respect to one delay parameter which is not associated with short-cut connection are obtained. Hopf bifurcations with respect to two other delays which are associated with short-cut connection are also obtained. Using the normal form method and center manifold theory, the direction of the Hopf bifurcation, stability and the properties of Hopf-bifurcating periodic solutions are determined. Using numerical simulations of the nonlinear model, different rich dynamical behaviors such as quasiperiodicity, torus attractor and chaotic-bands are also observed for suitable range of three delay parameters. Lyapunov exponents are also calculated using the AnT 4.669 tool for verification of chaotic dynamics.

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Acknowledgements

The authors are grateful to anonymous referees for their helpful suggestions to improve the manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah, 711103, India

    Amitava Kundu & Pritha Das

  2. Department of Mathematics, Jadavpur University, Jadavpur, India

    A. B. Roy

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  1. Amitava Kundu
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Kundu, A., Das, P. & Roy, A.B. Complex dynamics of a four neuron network model having a pair of short-cut connections with multiple delays. Nonlinear Dyn 72, 643–662 (2013). https://doi.org/10.1007/s11071-012-0742-2

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