About stability of equilibria of one system of stochastic delay differential equations with exponential nonlinearity
Pub. online: 13 February 2025
Type: Research Article
Open Access
Open Access
Received
12 October 2024
12 October 2024
Revised
1 February 2025
1 February 2025
Accepted
1 February 2025
1 February 2025
Published
13 February 2025
13 February 2025
Abstract
A system of two nonlinear delay differential equations under stochastic perturbations is considered. Nonlinearity of the exponential type in each equation of the system under consideration depends on the both variables of the system. The stability in probability of the zero and nonzero equilibria of the system is studied via the general method of Lyapunov functionals construction and the method of linear matrix inequalities (LMIs). The obtained results are illustrated via examples and figures with numerical simulations of solutions of a considered system of stochastic differential equations. The proposed way of investigation can be applied to nonlinear systems of higher dimension and with other types of nonlinearity, both for delay differential equations and for difference equations.
References
Beretta, E., Kolmanovskii, V., Shaikhet, L.: Stability of epidemic model with time delays influenced by stochastic perturbations. Math. Comput. Simul. 45(3-4), 269–277 (1998). Special Issue “Delay Systems”. MR1622405. https://doi.org/10.1016/S0378-4754(97)00106-7
Berezansky, L., Braverman, E., Idels, L.: Nicholson’s blowflies differential equations revisited: Main results and open problems. Appl. Math. Model. 34(6), 1405–1417 (2010). MR2592579. https://doi.org/10.1016/j.apm.2009.08.027
Boyd, S., El-Ghaoui, L., Feron, E., Balakrishnan, V.: Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, PA, USA (1994). MR1284712. https://doi.org/10.1137/1.9781611970777
Bradul, N., Shaikhet, L.: Stability of the positive point of equilibrium of Nicholson’s blowflies equation with stochastic perturbations: numerical analysis. Discrete Dyn. Nat. Soc. 25, 92959 (2007). MR2346516. https://doi.org/10.1155/2007/92959
Chen, W., Wang, W.: Global exponential stability for a delay differential neoclassical growth model. Adv. Differ. Equ. 2014(9), 325 (2014). MR3360574. https://doi.org/10.1186/1687-1847-2014-325
Choi, H.H.: A new method for variable structure control system design: A linear matrix inequality approach. Automatica 33, 2089–2092 (1997). MR1486914. https://doi.org/10.1016/S0005-1098(97)00118-0
Ding, X., Li, W.: Stability and bifurcation of numerical discretization Nicholson blowflies equation with delay. Discrete Dyn. Nat. Soc. 2006, 19413 (2006), 12 pp. MR2244293. https://doi.org/10.1155/DDNS/2006/19413
El-Metwally, E., Grove, E.A., Ladas, G., Levins, R., Radin, M.: On the difference equation ${x_{n+1}}=\alpha +\beta {x_{n-1}}{e^{-{x_{n}}}}$. Nonlinear Anal. 47(7), 4623–4634 (2001). MR1975856. https://doi.org/10.1016/S0362-546X(01)00575-2
Feckan, M., Marynets, K.: Study of differential equations with exponential nonlinearities via the lower and upper solutions’ method. Numer. Anal. Appl. Math. 1(2), 1–7 (2020). https://doi.org/10.36686/Ariviyal.NAAM.2020.01.02.007
Fridman, E., Shaikhet, L.: Stabilization by using artificial delays: an LMI approach. Automatica 81, 429–437 (2017). MR3654628. https://doi.org/10.1016/j.automatica.2017.04.015
Fridman, E., Shaikhet, L.: Simple LMIs for stability of stochastic systems with delay term given by Stieltjes integral or with stabilizing delay. Syst. Control Lett. 124, 83–91 (2019). MR3899077. https://doi.org/10.1016/j.sysconle.2018.12.007
Gikhman, I.I., Skorokhod, A.V.: Stochastic Differential Equations. Springer, Berlin, Germany (1972). MR0346904
Gouaisbaut, F., Dambrine, M., Richard, J.P.: Robust control of delay systems: a sliding mode control design via LMI. Syst. Control Lett. 46(4), 219–230 (2002). MR2010239. https://doi.org/10.1016/S0167-6911(01)00199-2
Kara, M.: Investigation of the global dynamics of two exponential-form difference equations systems. Electron. Res. Arch. 31(11), 6697–6724 (2023). MR4657209. https://doi.org/10.3934/era.2023338
Kolmanovskii, V., Shaikhet, L.: Some peculiarities of the general method of Lyapunov functionals construction. Appl. Math. Lett. 15(3), 355–360 (2002). MR1891559. https://doi.org/10.1016/S0893-9659(01)00143-4
Kolmanovskii, V., Shaikhet, L.: About one application of the general method of Lyapunov functionals construction. Int. J. Robust Nonlinear Control 13(9), 805–818 (2003). MR1998313. https://doi.org/10.1002/rnc.846
Li, J., Zhang, B., Li, Y.: Dependence of stability of Nicholson’s blowflies equation with maturation stage on parameters. J. Appl. Anal. Comput. 7(12), 670–680 (2017). MR3602445. https://doi.org/10.11948/2017042
Li, W.T., Fan, Y.H.: Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson’s blowflies model. J. Comput. Appl. Math. 201(1), 55–68 (2007). MR2293538. https://doi.org/10.1016/j.cam.2006.02.001
Nakamura, M., Ozawa, T.: Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth. Math. Z. 231, 479–487 (1999). MR1704989. https://doi.org/10.1007/PL00004737
Niculescu, S.I.: ${H^{\infty }}$ memory less control with an α-stability constraint for time delays systems: an LMI approach. IEEE Trans. Autom. Control 43(5), 739–743 (1998). MR1618043. https://doi.org/10.1109/9.668850
Papaschinopoulos, G., Ellina, G., Papadopoulos, K.B.: Asymptotic behavior of the positive solutions of an exponential type system of difference equations. Appl. Math. Comput. 245, 181–190 (2014). MR3260707. https://doi.org/10.1016/j.amc.2014.07.074
Papaschinopoulos, G., Fotiades, N., Schinas, C.J.: On a system of difference equations including negative exponential terms. J. Differ. Equ. Appl. 20(5-6), 717–732 (2014). MR3210311. https://doi.org/10.1080/10236198.2013.814647
Papaschinopoulos, G., Radin, M.A., Schinas, C.J.: On the system of two difference equations of exponential form: ${x_{n+1}}=a+b{x_{n-1}}{e^{-{y_{n}}}}$, ${y_{n+1}}=c+d{y_{n-1}}{e^{-{x_{n}}}}$. Math. Comput. Model. 54(11), 2969–2977 (2011). MR2841840. https://doi.org/10.1016/j.mcm.2011.07.019
Papaschinopoulos, G., Radin, M.A., Schinas, C.J.: Study of the asymptotic beha-vior of the solutions of three systems of difference equations of exponential form. Appl. Math. Comput. 218(9), 5310–5318 (2012). MR2870051. https://doi.org/10.1016/j.amc.2011.11.014
Seuret, A., Gouaisbaut, F.: Hierarchy of LMI conditions for the stability analysis of time-delay systems. Syst. Control Lett. 81, 1–7 (2015). MR3353689. https://doi.org/10.1016/j.sysconle.2015.03.007
Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Difference Equations. Springer Science & Business Media, London, UK (2011). MR3076210. https://doi.org/10.1007/978-3-319-00101-2
Shaikhet, L.: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations. Springer, Berlin, Germany (2013). MR3076210. https://doi.org/10.1007/978-3-319-00101-2
Shaikhet, L.: Stability of equilibrium states for a stochastically perturbed Mosquito popu-lation equation. Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms 21(2-3), 185–196 (2014). MR3298848
Shaikhet, L.: Stability of equilibrium states for a stochastically perturbed exponential type system of difference equations. J. Comput. Appl. Math. 290, 92–103 (2015). MR3370394. https://doi.org/10.1016/j.cam.2015.05.002
Shaikhet, L.: Stability of equilibriums of stochastically perturbed delay differential neoclassical growth model. Discrete Contin. Dyn. Syst., Ser. B 22(4), 1565–1573 (2017). MR3639178. https://doi.org/10.3934/dcdsb.2017075
Shaikhet, L.: Stability of the zero and positive equilibria of two connected neoclassical growth models under stochastic perturbations. Commun. Nonlinear Sci. Numer. Simul. 65, 86–93 (2019). Pub. online: August 2018. MR3860057. https://doi.org/10.1016/j.cnsns.2018.07.033
Shaikhet, L.: About stability of nonlinear stochastic differential equations with state-dependent delay. Symmetry 14(11), 2307 (2022). https://www.mdpi.com/2073-8994/14/11/2307/pdf
Shaikhet, L.: Stability of equilibria of exponential type system of three differential equations under stochastic perturbations. Math. Comput. Simul. 206, 105–117 (2023). MR4517038. https://doi.org/10.1016/j.matcom.2022.11.008
Shaikhet, L.: Stability of the exponential type system of stochastic difference equations. Mathematics 11(18), 3975 (2023). Special Issue “Nonlinear Stochastic Dynamics and Control and Its Applications”. https://www.mdpi.com/2227-7390/11/18/3975
Thai, T.H., Dai, N.A., Anh, P.T.: Global dynamics of some system of second-order difference equations. Electron. Res. Arch. 29(6), 4159–4175 (2021). MR4342296. https://doi.org/10.3934/era.2021077
Wan, H., Zhu, H.: A new model with delay for mosquito population dynamics. Math. Biosci. Eng. 11(6), 1395–1410 (2014). MR3263418. https://doi.org/10.3934/mbe.2014.11.1395
