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Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays

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Abstract

Time delay is an inevitable factor in neural networks due to the finite propagation velocity and switching speed. Neural system may lose its stability even for very small delay. In this paper, a two-neural network system with the different types of delays involved in self- and neighbor- connection has been investigated. The local asymptotic stability of the equilibrium point is studied by analyzing the corresponding characteristic equation. It is found that the multiple delays can lead the system dynamic behavior to exhibit stability switches. The delay-dependent stability regions are illustrated in the delay-parameter plane, followed which the double Hopf bifurcation points can be obtained from the intersection points of the first and second Hopf bifurcation, i.e., the corresponding characteristic equation has two pairs of imaginary eigenvalues. Taking the delays as the bifurcation parameters, the classification and bifurcation sets are obtained in terms of the central manifold reduction and normal form method. The dynamical behavior of system may exhibit the quasi-periodic solutions due to the Neimark- Sacker bifurcation. Finally, numerical simulations are made to verify the theoretical results.

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Acknowledgments

This research is supported by the State Key Program of National Natural Science of China under Grant No. 11032009, Shanghai Leading Academic Discipline Project in No. B302, the PhD Start-up Fund of Shanghai Ocean University in No. A-2400-11-0214 and Young Teacher Training Program of Colleges and Universities in Shanghai under Grant No. ZZhy12030.

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Authors and Affiliations

  1. School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, China

    Zi-Gen Song & Jian Xu

  2. College of Information Technology, Shanghai Ocean University, Shanghai, 201306, China

    Zi-Gen Song

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  1. Zi-Gen Song
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  2. Jian Xu
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Correspondence to Jian Xu.

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Song, ZG., Xu, J. Stability switches and double Hopf bifurcation in a two-neural network system with multiple delays. Cogn Neurodyn 7, 505–521 (2013). https://doi.org/10.1007/s11571-013-9254-0

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