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Graph-Theoretical Method to the Existence of Stationary Distribution of Stochastic Coupled Systems

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Abstract

In this paper, by linking Fokker–Planck equations with stochastic coupled systems, a new method is provided to investigate the existence of a stationary distribution of stochastic coupled systems. Based on the graph theory and the Lyapunov method, an appropriate Lyapunov function associated with stationary Fokker–Planck equations is constructed. Moreover, a Lyapunov-type theorem and a coefficients-type criterion are obtained to guarantee the existence of a stationary distribution. Furthermore, theoretical results are applied to explore the existence of a stationary distribution of stochastic predator–prey models with dispersal and a sufficient criterion is presented correspondingly. Finally, a numerical example is given to illustrate the effectiveness of our results.

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References

  1. Aghajani, A., Jalilian, Y., Roomi, V.: Oscillation theorems for the generalized Lienard system. Math. Comput. Model. 54, 2471–2478 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Fiedler, B., Belhaq, M., Houssni, M.: Basins of attraction in strongly damped coupled mechanical oscillators: a global example. Z. Angew. Math. Phys. 50, 282–300 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Florin, H., Sebastien, C., Patrick, C.: Robust synchronization of different coupled oscillators: application to antenna arrays. J. Frankl. Inst. Eng. Appl. Math. 346, 413–430 (2009)

    Article  MATH  Google Scholar 

  4. Jiang, W., Wei, J.: Bifurcation analysis in van der Pol’s oscillator with delayed feedback. J. Comput. Appl. Math. 213, 604–615 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Xiong, Y., Xing, J., Price, W.: Power flow analysis of complex coupled systems by progressive approaches. J. Sound Vibr. 239, 275–295 (2001)

    Article  Google Scholar 

  6. Xiong, Y., Xing, J., Price, W.: A general linear mathematical model of power flow analysis and control for integrated structure–control systems. J. Sound Vibr. 267, 301–334 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ball, F., Sirl, D., Trapman, P.: Analysis of a stochastic SIR epidemic on a random network incorporating household structure. Math. Biosci. 224, 53–73 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Shu, H., Fan, D., Wei, J.: Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission. Nonlinear Anal. 13, 1581–1592 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Florian, D., Francesco, B.: Synchronization in complex networks of phase oscillators: a survey. Automatica 50, 1539–1564 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hu, L., Mao, X.: Almost sure exponential stabilisation of stochastic systems by state-feedback control. Automatica 44, 465–471 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Li, C., Chen, L., Aihara, K.: Stochastic synchronization of genetic oscillator networks. BMC Syst. Biol. 1, 1–12 (2007)

    Article  Google Scholar 

  12. Ghorbanian, P., Ramakrishnan, S., Whitman, A., Ashrafiuon, H.: A phenomenological model of EEG based on the dynamics of a stochastic Duffing-van der Pol oscillator network. Biomed. Signal Process. Control 15, 1–10 (2015)

    Article  Google Scholar 

  13. Rosas, A., Escaff, D., Pinto, I., Lindenberg, K.: Globally coupled stochastic two-state oscillators: synchronization of infinite and finite arrays. J. Phys. A 49, 1–24 (2016)

    MathSciNet  MATH  Google Scholar 

  14. Moradi, S., Anderson, J., Gürcan, O.D.: Predator-prey model for the self-organization of stochastic oscillators in dual populations. Phys. Rev. 92(6), 062930 (2015)

    MathSciNet  Google Scholar 

  15. Kao, Y., Wang, C., Karimi, H.R., Bi, R.: Global stability of coupled Markovian switching reaction–diffusion systems on networks. Nonlinear Anal. 13, 61–73 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Li, W., Song, H., Qu, Y., Wang, K.: Global exponential stability for stochastic coupled systems on networks with Markovian switching. Syst. Control Lett. 62, 468–474 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cui, J., Li, Q., Hu, G., Tao, Z., Lu, Z.: Asymptotical stability for 2-D stochastic coupled FMII models on networks. Int. J. Control Autom. Syst. 13, 1550–1555 (2015)

    Article  Google Scholar 

  18. Zhang, C., Li, W., Su, H., Wang, K.: Asymptotic boundedness for stochastic coupled systems on networks with Markovian switching. Neurocomputing 136, 180–189 (2014)

    Article  Google Scholar 

  19. Zhang, C., Li, W., Su, H., Wang, K.: A graph-theoretic approach to boundedness of stochastic Cohen–Grossberg neural networks with Markovian switching. Appl. Math. Comput. 219, 9165–9173 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Zhang, C., Li, W., Wang, K.: Boundedness for network of stochastic coupled van der Pol oscillators with time-varying delayed coupling. Appl. Math. Model. 37, 5394–5402 (2013)

    Article  MathSciNet  Google Scholar 

  21. Tan, J., Li, C.: Global synchronization of discrete-time coupled neural networks with Markovian switching and impulses. Int. J. Biomath. 9, 1650041 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, C., Li, W., Wang, K.: Graph-theoretic method on exponential synchronization of stochastic coupled networks with Markovian switching. Nonlinear Anal. 15, 37–51 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Wu, E., Yang, X.: Adaptive synchronization of coupled nonidentical chaotic systems with complex variables and stochastic perturbations. Nonlinear Dyn. 84, 261–269 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Mao, X., Yuan, C., Yin, G.: Numerical method for stationary distribution of stochastic differential equations with Markovian switching. J. Comput. Appl. Math. 174, 1–27 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Huang, W., Ji, M., Liu, Z., Yi, Y.: Steady states of Fokker–Planck. J. Dyn. Differ. Equ. 27, 721–742 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Huang, W., Ji, M., Liu, Z., Yi, Y.: Steady states of Fokker–Planck equations: II. Non-existence. J. Dyn. Differ. Equ. 27, 743–762 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Huang, W., Ji, M., Liu, Z., Yi, Y.: Steady states of Fokker–Planck equations: III. Degenerate diffusion. J. Dyn. Differ. Equ. 28, 127–141 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Herzog, D.P., Mattingly, J.C.: Noise-induced stabilization of planar flows I. Electron. J. Probab. 20, 1–43 (2015)

    MathSciNet  MATH  Google Scholar 

  29. Athreya, A., Kolba, T., Mattingly, J.C.: Propagating Lyapunov functions to prove noise-induced stabilization. Electron. J. Probab. 17, 1–38 (2012)

    MathSciNet  MATH  Google Scholar 

  30. Li, M.Y., Shuai, Z.: Global-stability problem for coupled systems of differential equations on networks. J. Differ. Equ. 248, 1–20 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  31. Guo, H., Li, M.Y., Shuai, Z.: A graph-theoretic approach to the method of global Lyapunov functions. Proc. Am. Math. Soc. 136, 2793–2802 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Mao, X.: Stochastic Differential Equations and Applications. Horwood, Chichester (1997)

    MATH  Google Scholar 

  33. Zhao, Y., Yuan, S., Zhang, T.: The stationary distribution and ergodicity of a stochastic phytoplankton allelopathy model under regime switching. Commun. Nonlinear Sci. Numer. Simul. 37, 131–142 (2016)

    Article  MathSciNet  Google Scholar 

  34. Chen, F., Huang, A.: On a nonautonomous predator–prey model with prey dispersal. Appl. Math. Comput. 184, 809–822 (2007)

    MathSciNet  MATH  Google Scholar 

  35. Kuang, Y., Takeuchi, Y.: Predator–prey dynamics in models of prey dispersal in two-patch environments. Math. Biosci. 120, 77–98 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  36. Xu, C., Tang, X., Liao, M.: Stability and bifurcation analysis of a delayed predator–prey model of prey dispersal in two-patch environments. Appl. Math. Comput. 216, 2920–2936 (2010)

    MathSciNet  MATH  Google Scholar 

  37. Xu, R., Ma, Z.: The effect of dispersal on the permanence of a predator–prey system with time delay. Nonlinear Anal. RWA 9, 354–369 (2008)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors really appreciate the reviewer’s valuable comments. The second author was supported by the NNSF of China (Nos. 11301112 and 11401136), the NNSF of Shandong Province (Nos. ZR2013AQ003 and ZR2014AQ010) and China Postdoctoral Science Foundation funded Project (No. 2014T70313). The third author was supported by the National Natural Science Foundation of China (61401283) and Educational Commission of Guangdong Province, China (2014KTSCX113).

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Authors and Affiliations

  1. Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai, 264209, People’s Republic of China

    Yan Liu & Wenxue Li

  2. College of mathematics and statistics, Shenzhen University, Shenzhen, 518060, People’s Republic of China

    Jiqiang Feng

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  1. Yan Liu
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  2. Wenxue Li
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  3. Jiqiang Feng
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Correspondence to Wenxue Li.

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Liu, Y., Li, W. & Feng, J. Graph-Theoretical Method to the Existence of Stationary Distribution of Stochastic Coupled Systems. J Dyn Diff Equat 30, 667–685 (2018). https://doi.org/10.1007/s10884-016-9566-y

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  • DOI: https://doi.org/10.1007/s10884-016-9566-y

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