Stability analysis and absolute synchronization of a three-unit delayed neural network

Wang, Lin Jun ; Xie, You Xiang ; Wei, Zhou Chao ; Peng, Jian

Kybernetika, Tome 51 (2015) no. 5, pp. 800-813

Abstract (VO)

In this paper, we consider a three-unit delayed neural network system, investigate the linear stability, and obtain some sufficient conditions ensuring the absolute synchronization of the system by the Lyapunov function. Numerical simulations show that the theoretically predicted results are in excellent agreement with the numerically observed behavior.

Détail

DOI : 10.14736/kyb-2015-5-0800

Classification : 34D06, 34D20
Keywords: absolute synchronization; delay; linear stability; neural network

Bibliographie

[1] Albertini, F., Alessandro, D.: Further conditions on the stability of continuous time systems with saturation. IEEE Trans. Circuits Syst I. 47 (2000), 723-729. | DOI

[2] Bélair, J., Campbell, S. A., Driessche, P. van den: Frustration, stability, and delay-induced oscillations in a neural network model. SIAM J. Appl. Math. 56 (1996), 245-255. | DOI | MR

[3] Bélair, J.: Stability in a model of a delayed neural network. J. Dynam. Dif. Equns. 5 (1993), 603-623. | DOI | MR | Zbl

[4] Bélair, J., Campbell, S. A.: Stability and bifurcations of equilibria in a multiple-delayed differential equation. SIAM J. Appl. Math. 54 (1994), 1402-1424. | DOI | MR | Zbl

[5] Beuter, A., Belair, J., Labrie, C.: Feedback and delays in neurological diseases: a modelling study using dynamical systems. Bull. Math. Biol. 55 (1993), 525-541. | DOI

[6] Campbell, S. A., Ruan, S., Wolkowicz, G. S. K., Wu, J.: Stability and bifurcation of a simple neural network with multiple time delays. Fields Institute Communications, vol. 21, American Mathematical Society, Providence, RI, 1998, pp. 65-79. | MR

[7] Chen, Y., Huang, Y., Wu, J.: desynchronization of large scale delayed neural networks. Proc. Amer. Math. Soc. 128 (2000), 2365-2371. | DOI | MR | Zbl

[8] Chen, Y., Lü, J. H., Lin, Z. L.: Consensus of discrete-time multi-agent systems with transmission nonlinearity. Automatica 49 (2013), 1768-1775. | DOI | MR

[9] Cushing, J. M.: Integro-differential Equations and Delay Models in Population Dynamics. Lecture Notes in Biomath, vol. 20, Springer, New York 1977. | DOI

[10] Dhamala, M., Jirsa, V., Ding, M.: Enhancement of neural synchrony by time delay. Phys. Rev. Lett. 92 (2004), 74-104. | DOI

[11] Diekmann, O., Gils, S. A. van, Lunel, S. M. Verduyn, Walther, H. O.: Delay Equations, Functional, Complex, and Nonlinear Analysis. Springer Verlag, New York 1995. | DOI | MR

[12] ., P., Driessche, Zou, X.: Global attractivity in delayed Hopfield neural network models. SIAM J. Appl. Math. 58 (1998), 1878-1890. | DOI | MR

[13] Faria, T.: On a planar system modelling a neuron network with memory. J. Diff. Equns. 168 (2000), 129-149. | DOI | MR | Zbl

[14] Faria, T., Magalhes, L. T.: Normal forms for retarded functional differential equations with parameters and applications to Hopf bifurcation. J. Diff. Equations. 122 (1995), 181-200. | DOI | MR

[15] Gopalsamy, K., Leung, I.: Delay induced periodicity in a neural network of excitation and inhibition. Physica D. 89 (1996), 395-426. | DOI | MR

[16] Hale, J.: Theory of Functional Differential Equations. Springer, New York 1997. | MR | Zbl

[17] Hale, J., Kocak, H.: Dynamics and Bifurcations. Springer, New York 1991. | MR | Zbl

[18] Hale, J., Lunel, S. V.: Introduction to Functional Differential Equations. Springer, New York 1993. | DOI | MR | Zbl

[19] Hirsch, M. W.: Convergent activation dynamics in continuous-time networks. Neural Networks 2 (1989), 331-349. | DOI

[20] Hopfield, J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. USA 81 (1994), 3088-3092. | DOI

[21] Huang, L., Wu, J.: Dynamics of inhibitory artificial neural networks with threshold nonlinearity. Fields Ins. Commun. 29 (2001), 235-243. | MR | Zbl

[22] Karimi, H. R., Gao, H. J.: New Delay-Dependent Exponential H$\infty$ Synchronization for Uncertain Neural Networks with Mixed Time Delays. IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics 40 (2010), 173-185. | DOI

[23] Liu, Y. R., Wang, Z. D., Liang, J. L.: Stability and synchronization of discrete-time Markovian jumping neural networks with mixed mode-dependent time delays. IEEE Transactions on Neural Networks 20 (2009), 1102-1116. | DOI

[24] Lü, J. H., Chen, G. R.: A time-varying complex dynamical network model and its controlled synchronization criteria. IEEE Trans. Automat. Control 50 (2005), 6, 841-846. | DOI | MR

[25] Marcus, C. M., Westervelt, R. M.: Stability of analog neural networks with delay. Phys. Rev. A. 39 (1989), 347-359. | DOI | MR

[26] Niebur, E., Schuster, H., Kammen, D.: Collective frequencies and meta stability in networks of limit-cycle oscillators with time delay. Phys. Rev. Lett. 67 (1991), 2753-2756. | DOI

[27] Rosenblum, M. G., Pikovsky, A. S.: Controlling synchronization in an ensemble of globally coupled oscillators. Phys. Rev. Lett. 92 (2004), 102-114. | DOI

[28] Ruan, S., Wei, J.: On the zeros of transcendental functions with applications to stability of delayed differential equations with two delays, Dyn. Discrete Impuls Syst Ser A: Math. Anal. 10 (2003), 63-74. | MR

[29] Wu, J.: Introduction to Neural Dynamics and Signal Transmission Delay. Walter de Cruyter, Berlin 2001. | DOI | MR | Zbl

[30] Wu, J.: Symmetric functional differential equations and neural networks with memory. Trans. Amer. Math. Soc. 350 (1998), 4799-4838. | DOI | MR | Zbl

[31] Yeung, M., Strogatz, S.: Time delay in the Kuramoto model of coupled oscillators. Phys. Rev. Lett. 82 (1999), 648-651. | DOI | MR

[32] Zhou, J., Lu, J. A., Lü, J. H.: Adaptive synchronization of an uncertain complex dynamical network. IEEE Trans. Automat. Control 51 (2006), 4, 652-656. | DOI | MR