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:
[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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.jde.2009.09.003.

[10]

S. Elaydi, An Introduction to Difference Equations,, 3rd ed, (2004).

[11]

G. Barlev, M. Girvan and E. Ott, Map model for synchronization of systems of many coupled oscillators,, Chaos, 20 (2010). 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. 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. 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). 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). 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). 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). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/j.jde.2009.09.003.

[10]

S. Elaydi, An Introduction to Difference Equations,, 3rd ed, (2004).

[11]

G. Barlev, M. Girvan and E. Ott, Map model for synchronization of systems of many coupled oscillators,, Chaos, 20 (2010). 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. 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. 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). 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). 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). 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). 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. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.1016/S0167-2789(03)00285-9.

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