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Relaxation oscillation and attractive basins of a two-neuron Hopfield network with slow and fast variables

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Abstract

In this paper, the nonlinear dynamics of a two-neuron Hopfield network with slow and fast variables is investigated. By means of the geometric singular perturbation theory, the condition that ensures the existence of the relaxation oscillation is obtained, the period of the relaxation oscillation is determined analytically, and the shape of attractive basin of every stable equilibrium is figured out. By using the method of stability switches, the delay effect on the characteristics of the relaxation oscillation and the attractive basins is studied. Case studies are given to demonstrate the validity of theoretical analysis.

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Acknowledgements

The authors are grateful to the financial support of NSF of China under Grants 11032009, 11102078, and JXNSF of China CA201107114.

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Authors and Affiliations

  1. School of Mathematics and Information Science, Nanchang Hangkong University, 330063, Nanchang, Jiangxi, China

    Y. G. Zheng

  2. Institute of Science, PLA University of Science and Technology, 211101, Nanjing, China

    Z. H. Wang

  3. State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, 210016, Nanjing, China

    Z. H. Wang

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  1. Y. G. Zheng
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Correspondence to Z. H. Wang.

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Zheng, Y.G., Wang, Z.H. Relaxation oscillation and attractive basins of a two-neuron Hopfield network with slow and fast variables. Nonlinear Dyn 70, 1231–1240 (2012). https://doi.org/10.1007/s11071-012-0527-7

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  • DOI: https://doi.org/10.1007/s11071-012-0527-7

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