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

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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

Huan Su ^1, , Pengfei Wang ^1, and Xiaohua Ding ^1,

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

Received January 2015 Revised July 2015 Published November 2015

Abstract
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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.

Keywords: Global stability, multi-diffusion, graph-theoretic method, nonstandard finite difference scheme..

Mathematics Subject Classification: Primary: 39A30; Secondary: 37J25, 92B9.

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 and Continuous Dynamical Systems - B, 2016, 21 (1) : 253-269. doi: 10.3934/dcdsb.2016.21.253

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	S. Elaydi, An Introduction to Difference Equations, 3rd ed, (Springer, New York, 2004).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996.
[26]	R. E. Mickens, Nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl., 8 (2002), 823-847. doi: 10.1080/1023619021000000807.
[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.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[10]	S. Elaydi, An Introduction to Difference Equations, 3rd ed, (Springer, New York, 2004).
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[25]	D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, 1996.
[26]	R. E. Mickens, Nonstandard finite difference schemes for differential equations, J. Differ. Equ. Appl., 8 (2002), 823-847. doi: 10.1080/1023619021000000807.
[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.

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