# American Institute of Mathematical Sciences

January  2016, 21(1): 253-269. doi: 10.3934/dcdsb.2016.21.253

## Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application

 1 Department of Mathematics, Harbin Institute of Technology at Weihai, Weihai 264209, China, China, China

Received  January 2015 Revised  July 2015 Published  November 2015

In this paper, we investigate the global stability of discrete-time coupled systems with multi-diffusion (DCSMDs). By utilizing a multi-digraph theory, we construct a global Lyapunov function for DCSMDs. Consequently, some sufficient conditions are presented to ensure the stability of a general DCSMDs. Then the proposed theory is successfully applied to analyze the global stability for a discrete-time predator-prey model which is discretized by a nonstandard finite difference scheme. Finally, an example with numerical simulation is given to demonstrate the effectiveness of the obtained results.
Citation: Huan Su, Pengfei Wang, Xiaohua Ding. Stability analysis for discrete-time coupled systems with multi-diffusion by graph-theoretic approach and its application. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 253-269. doi: 10.3934/dcdsb.2016.21.253
##### References:
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Wang, Global stability of coupled nonlinear systems with Markovian switching, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2609-2616. doi: 10.1016/j.cnsns.2011.09.039.  Google Scholar [24] C. Zhang, W. Li and K. Wang, Graph-theoretic approach to stability of multi-group models with dispersal, Discrete. Cont. Dyn-B., 20 (2015), 259-280. doi: 10.3934/dcdsb.2015.20.259.  Google Scholar [25] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996.  Google Scholar [26] R. E. Mickens, Nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl., 8 (2002), 823-847. doi: 10.1080/1023619021000000807.  Google Scholar [27] S. M. Moghadas, M. E. Alexander and B. D. Corbett, A non-standard numerical scheme for a generalized Gause-type predator-prey model, Physica D., 188 (2004), 134-151. doi: 10.1016/S0167-2789(03)00285-9.  Google Scholar

show all references

##### References:
 [1] J. Wang, J. Zu, X. Liu, G. Huang and J. Zhang, Global dynamics of a multi-group epidemic model with general relapse distribution and nonlinear incidence rate, J. Biol. Syst., 20 (2012), 235-258. doi: 10.1142/S021833901250009X.  Google Scholar [2] H. Shu, D. Fan and J. Wei, Global stability of multi-group SEIR epidemic models with distributed delays and nonlinear transmission, Nonlinear Anal. RWA, 13 (2012), 1581-1592. doi: 10.1016/j.nonrwa.2011.11.016.  Google Scholar [3] C. Ji, D. Jiang, Q. Yang and N. Shi, Dynamics of a multigroup SIR epidemic model with stochastic perturbation, Automatica, 48 (2012), 121-131. doi: 10.1016/j.automatica.2011.09.044.  Google Scholar [4] H. Chen and J. Sun, Global stability of delay multigroup epidemic models with group mixing and nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 4391-4400. doi: 10.1016/j.amc.2011.10.015.  Google Scholar [5] R. Sun and J. Shi, Global stability of multigroup epidemic model with group mixing and nonlinear incidence rates, Appl. Math. Comput., 218 (2011), 280-286. doi: 10.1016/j.amc.2011.05.056.  Google Scholar [6] M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361 (2010), 38-47. doi: 10.1016/j.jmaa.2009.09.017.  Google Scholar [7] Y. Muroya, Y. Enatsu and T. Kuniya, Global stability for a multi-group SIRS epidemic model with varying population sizes, Nonlinear Anal. RWA, 14 (2013), 1693-1704. doi: 10.1016/j.nonrwa.2012.11.005.  Google Scholar [8] J. Epperlein, S. Siegmund and P. Stehík, Evolutionary games on graphs and discrete dynamical systems， J. Difference Eq. Appl., 21 (2015), 72-95. doi: 10.1080/10236198.2014.988618.  Google Scholar [9] M. Y. Li and Z. Shuai, Global-stability problem for coupled systems of differential equations on networks, J. Differ. Equ., 248 (2010), 1-20. doi: 10.1016/j.jde.2009.09.003.  Google Scholar [10] S. Elaydi, An Introduction to Difference Equations, 3rd ed, (Springer, New York, 2004). Google Scholar [11] G. Barlev, M. Girvan and E. Ott, Map model for synchronization of systems of many coupled oscillators, Chaos, 20 (2010), 023109. doi: 10.1063/1.3357983.  Google Scholar [12] M. Lazar, W. P. M. H. Heemels and A. R. Teel, Lyapunov functions, stability and input-to-state stability subtleties for discrete-time discontinuous systems, IEEE Trans. Autom. Control., 54 (2009), 2421-2425. doi: 10.1109/TAC.2009.2029297.  Google Scholar [13] J. Q. Qiu, K. F. Lu, P. Shi and M. S. Mahmoud, Robust exponential stability for discrete-time interval BAM neural networks with delays and Markovian jump parameters, Int. J. Adapt. Control., 24 (2010), 760-785. doi: 10.1002/acs.1171.  Google Scholar [14] M. S. Peng and X. Z. Yang, New stability criteria and bifurcation analysis for nonlinear discrete-time coupled loops with multiple delays, Chaos, 20 (2010), 013125, 11pp. doi: 10.1063/1.3339857.  Google Scholar [15] S. V. Naghavi and A. A. Safavi, Novel synchronization of discrete-time chaotic systems using neural network observer, Chaos, 18 (2008), 033110, 9pp. doi: 10.1063/1.2959140.  Google Scholar [16] J. D. Cao and J. Q. Lu, Adaptive synchronization of neural networks with or without time-varying delay, Chaos, 16 (2006), 013133, 6pp. doi: 10.1063/1.2178448.  Google Scholar [17] H. Su, W. Li and K. Wang, Global stability of discrete-time coupled systems on networks and its applications, Chaos, 22 (2012), 033135, 11pp. doi: 10.1063/1.4748851.  Google Scholar [18] C. Zhang, W. Li and K. Wang, Graph theory-based approach for stability analysis of stochastic coupled systems with Lévy noise on networks, IEEE Trans. Neural Netw. Learn. Syst., 26 (2014), 1698-1709. doi: 10.1109/TNNLS.2014.2352217.  Google Scholar [19] F. M. Atay and $\ddot Q$. Karabacak, Stability of coupled map networks with delays, SIAM J. Appl. Dyn. Syst., 5 (2006), 508-527. doi: 10.1137/060652531.  Google Scholar [20] H. Guo, M. L. Li and Z.Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793-2802. doi: 10.1090/S0002-9939-08-09341-6.  Google Scholar [21] C. Zhang, W. Li and K. Wang, Boundedness for network of stochastic coupled van der Pol oscillators with time-varying delayed coupling, Appl. Math. Model., 37 (2013), 5394-5402. doi: 10.1016/j.apm.2012.10.032.  Google Scholar [22] C. Zhang, W. Li and K. Wang, A graph-theoretic approach to stability of neutral stochastic coupled oscillators network with time-varying delayed coupling, Math. Meth. Appl. Sci., 37 (2014), 1179-1190. doi: 10.1002/mma.2879.  Google Scholar [23] W. Li, H. Su, D. Wei and K. Wang, Global stability of coupled nonlinear systems with Markovian switching, Commun. Nonlinear Sci. Numer. Simulat., 17 (2012), 2609-2616. doi: 10.1016/j.cnsns.2011.09.039.  Google Scholar [24] C. Zhang, W. Li and K. Wang, Graph-theoretic approach to stability of multi-group models with dispersal, Discrete. Cont. Dyn-B., 20 (2015), 259-280. doi: 10.3934/dcdsb.2015.20.259.  Google Scholar [25] D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996.  Google Scholar [26] R. E. Mickens, Nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl., 8 (2002), 823-847. doi: 10.1080/1023619021000000807.  Google Scholar [27] S. M. Moghadas, M. E. Alexander and B. D. Corbett, A non-standard numerical scheme for a generalized Gause-type predator-prey model, Physica D., 188 (2004), 134-151. doi: 10.1016/S0167-2789(03)00285-9.  Google Scholar
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