On the periodic solution of a class of stochastic nonlinear system with delays

Du, Bo ¹ ; Wang, Haiyan ² ; Liu, Maoxing ³ ; Cheng, Xiwang ¹

¹ Department of Mathematics, Huaiyin Normal University, Huaian Jiangsu, 223300, P. R. China
² School of Mathematical and Natural Sciences, Arizona State University, Arizona, U. S. A.
³ Department of Mathematics, North University of China, Taiyuan 030051, P. R. China

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 2, p. 263-273.

Résumé

This paper is devoted to investigating a class of stochastic nonlinear system with periodic coefficients. Some criteria on existence and uniqueness of periodic solution are established for the stochastic nonlinear system. Finally, a numerical example is given to show the effectiveness and merits of the present results.

Détail

DOI : 10.22436/jnsa.011.02.08

Classification : 34K50
Keywords: Periodic solution, stochastic, Itô's formula, existence

Bibliographie

[1] R. Z. Has’minskii On the dissipativity of random processes defined by differential equations, Problems of Information Transmission, Volume 1 (1965), pp. 88-104

[2] R. Z. Has’minskii Stochastic Stability of Differential Equations, Sijthoff and Noordhoff , Springer Science & Business Media, NetherIands, 1980

[3] K. Itô On Stochastic Differential Equations, Mem. Amer. Math. Soc., American Mathematical Society, Rhode Island, 1951

[4] Itô, K.; Nisio, M. On stationary solutions of a stochastic differential equation, J. Math. Kyoto Univ., Volume 4 (1964), pp. 1-75 | DOI

[5] Jiang, D.; Shi, N. A note on nonautonomous logistic equation with random perturbation, J. Math. Anal. Appl., Volume 303 (2005), pp. 164-172 | Zbl | DOI

[6] Jiang, D.; Shi, N.; Li, X. Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation, J. Math. Anal. Appl., Volume 340 (2008), pp. 588-597 | Zbl | DOI

[7] Kolmanovskii, V.; Myshkis, A. Introduction to the Theory and Application of Functional Differential Equations, Kluwer Academic Publishers, Dordrecht, 1999 | DOI

[8] Li, C.; Chen, L.; K. Aihara Impulsive control of stochastic systems with applications in chaos control, chaos synchronization, and neural networks, Chaos, Volume 2008 (2008), pp. 1-11 | Zbl | DOI

[9] Li, D.; D. Xu Periodic solutions of stochastic delay differential equations and applications to Logistic equation and neural networks, J. Korean Math. Soc., Volume 50 (2013), pp. 1165-1181 | Zbl | DOI

[10] Liang, J.; Wang, Z.; Liu, Y.; X. Liu Robust synchronization of an array of coupled stochastic discrete-time delayed neural networks, IEEE Trans. Neural Netw., Volume 19 (2008), pp. 1910-1921 | DOI

[11] Lu, K.; B. Schmalfuss Invariant manifolds for stochastic wave equations, J. Differential Equations, Volume 236 (2007), pp. 460-492 | DOI

[12] X. R. Mao Exponential Stabiity of Stochastic Differential Equations, Marcel Dekker, New York, 1994

[13] B. Oksendal Stochastic Differential Equations: An Introduction with Applications (Sixth Edition), Springer-Verlag, Berlin, 2003 | Zbl | DOI

[14] Tang, Y.; W. K.Wong Distributed synchronization of coupled neural networks via randomly occurring control , IEEE Trans. Neural Netw. Learn. Syst., Volume 24 (2013), pp. 435-447 | DOI

[15] Tojtovska, B.; S. Janković On some stability problems of impulsive stochastic Cohen-Grossberg neural networks with mixed time delays, Appl. Math. Comput., Volume 239 (2014), pp. 211-226 | Zbl | DOI

[16] Wu, X.; Tang, Y.; Zhang, W. Stability analysis of switched stochastic neural networks with time-varying delays, Neural Network, Volume 52 (2014), pp. 39-49 | Zbl | DOI

[17] Xu, D.; Huang, Y.; Z. Yang Existence theorems for periodic Markov process and stochastic functional differential equations, Discrete Contin. Dyn. Syst., Volume 24 (2009), pp. 1005-1023 | Zbl

[18] Zhang, Z.; Cao, J.; Do. Zhou Novel LMI-Based condition on global asymptotic stability for a class of Cohen-Grossberg BAM networks with extended activation functions , IEEE Transcation on Neural Network and learning system, Volume 25 (2014), pp. 1161-1172 | DOI

[19] Zhang, B. G.; K. Gopalsamy On the periodic solution of n-dimensional stochastic population models, Stochastic Anal. Appl., Volume 18 (2000), pp. 323-331 | DOI | Zbl

[20] Zhang, Z.; K. Liu Existence and global exponential stability of a periodic solution to interval general bidirectional associative memory (BAM) neural networks with multiple delays on time scales , Neural Network, Volume 24 (2011), pp. 427-439 | DOI | Zbl

[21] Zhang, W.; Tang, Y.; Wong, W.; Q. Miao Stochastic stability of delayed neural networks with Local impulsive effects, IEEE Trans. Neural Netw. Learn. Syst., Volume 26 (2015), pp. 2336-2345 | DOI

[22] Zhu, Q.; J. Cao Exponential stability of stochastic neural networks with both Markovian jump parameters and mixed time delays, IEEE Trans. Syst., Man, Cybern. B, Cybern., Volume 41 (2011), pp. 341-353 | DOI

[23] Zhu, Q.; J. Cao Stability of Markovian jump neural networks with impulse control and time varying delays, Nonlinear Anal. Real World Appl., Volume 13 (2012), pp. 2259-2270 | DOI | Zbl

[24] Zhou, H.; Zhou, Z.; Qiao, Z. Mean-square almost periodic solution for impulsive stochastic Nicholson’s blowflies model with delays, Appl. Math. Comput., Volume 219 (2013), pp. 5943-5948 | Zbl | DOI