    • Previous Article
The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting
• DCDS-B Home
• This Issue
• Next Article
Weak pullback attractors for stochastic Ginzburg-Landau equations in Bochner spaces

## Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise

 Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

* Corresponding author: Xiang Lv

Received  October 2020 Revised  March 2021 Published  April 2021

Fund Project: This work was supported by the National Natural Science Foundation of China (NSFC) under Grants No.11501369, No.11771295 and No.11971316; the NSF of Shanghai under Grants No.19ZR1437100 and No. 20JC1413800; Chen Guang Project(14CG43) of Shanghai Municipal Education Commission, Shanghai Education Development Foundation; Yangfan Program of Shanghai (14YF1409100) and Shanghai Gaofeng Project for University Academic Program Development

This paper is concerned with the existence of unstable stationary solutions for nonlinear stochastic differential equations (SDEs) with additive white noise. Assume that the nonlinear term $f$ is monotone (or anti-monotone) and the global Lipschitz constant of $f$ is smaller than the positive real part of the principal eigenvalue of the competitive matrix $A$, the random dynamical system (RDS) generated by SDEs has an unstable $\mathscr{F}_+$-measurable random equilibrium, which produces a stationary solution for nonlinear SDEs. Here, $\mathscr{F}_+ = \sigma\{\omega\mapsto W_t(\omega):t\geq0\}$ is the future $\sigma$-algebra. In addition, we get that the $\alpha$-limit set of all pull-back trajectories starting at the initial value $x(0) = x\in\mathbb{R}^n$ is a single point for all $\omega\in\Omega$, i.e., the unstable $\mathscr{F}_+$-measurable random equilibrium. Applications to stochastic neural network models are given.

Citation: Xiang Lv. Existence of unstable stationary solutions for nonlinear stochastic differential equations with additive white noise. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021133
##### References:
  L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar  I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, vol. 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar  J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. U.S.A., 79 (1982), 2554-2558.  doi: 10.1073/pnas.79.8.2554.  Google Scholar  J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci., 81 (1984), 3088-3092.  doi: 10.1073/pnas.81.10.3088. Google Scholar  J. Jiang and X. Lv, A small-gain theorem for nonlinear stochastic systems with inputs and outputs I: Additive white noise, SIAM J. Control Optim., 54 (2016), 2383-2402.  doi: 10.1137/15M1044047.  Google Scholar  J. Jiang and X. Lv, Global stability of feedback systems with multiplicative noise on the nonnegative orthant, SIAM J. Control Optim., 56 (2018), 2218-2247.  doi: 10.1137/16M1101076.  Google Scholar  R. Kha'sminskii, Stochastic Stability of Differential Equations, Alphen: Sijtjoff and Noordhoff, 1980. Google Scholar  H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1990. Google Scholar  X. Li and X. Mao, Stabilisation of highly nonlinear hybrid stochastic differential delay equations by delay feedback control, Automatica, 112 (2020), 108657. doi: 10.1016/j.automatica.2019.108657.  Google Scholar  X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. Google Scholar  X. Mao, Stochastic versions of the LaSalle theorem, J. Differential Equations, 153 (1999), 175-195.  doi: 10.1006/jdeq.1998.3552.  Google Scholar  X. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.  Google Scholar  M. Marcondes de Freitas and E. D. Sontag, A small-gain theorem for random dynamical systems with inputs and outputs, SIAM J. Control Optim., 53 (2015), 2657-2695.  doi: 10.1137/140991340.  Google Scholar  B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 5th ed., Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03620-4.  Google Scholar  H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995. Google Scholar  P. van den Driessche and X. Zou, Global attractivity in delayed Hopfield neural network models, SIAM J. Appl. Math., 58 (1998), 1878-1890.  doi: 10.1137/S0036139997321219.  Google Scholar  J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2.  Google Scholar

show all references

##### References:
  L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar  I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, vol. 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar  J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities, Proc. Nat. Acad. Sci. U.S.A., 79 (1982), 2554-2558.  doi: 10.1073/pnas.79.8.2554.  Google Scholar  J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-stage neurons, Proc. Nat. Acad. Sci., 81 (1984), 3088-3092.  doi: 10.1073/pnas.81.10.3088. Google Scholar  J. Jiang and X. Lv, A small-gain theorem for nonlinear stochastic systems with inputs and outputs I: Additive white noise, SIAM J. Control Optim., 54 (2016), 2383-2402.  doi: 10.1137/15M1044047.  Google Scholar  J. Jiang and X. Lv, Global stability of feedback systems with multiplicative noise on the nonnegative orthant, SIAM J. Control Optim., 56 (2018), 2218-2247.  doi: 10.1137/16M1101076.  Google Scholar  R. Kha'sminskii, Stochastic Stability of Differential Equations, Alphen: Sijtjoff and Noordhoff, 1980. Google Scholar  H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1990. Google Scholar  X. Li and X. Mao, Stabilisation of highly nonlinear hybrid stochastic differential delay equations by delay feedback control, Automatica, 112 (2020), 108657. doi: 10.1016/j.automatica.2019.108657.  Google Scholar  X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997. Google Scholar  X. Mao, Stochastic versions of the LaSalle theorem, J. Differential Equations, 153 (1999), 175-195.  doi: 10.1006/jdeq.1998.3552.  Google Scholar  X. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.  Google Scholar  M. Marcondes de Freitas and E. D. Sontag, A small-gain theorem for random dynamical systems with inputs and outputs, SIAM J. Control Optim., 53 (2015), 2657-2695.  doi: 10.1137/140991340.  Google Scholar  B. Øksendal, Stochastic Differential Equations: An Introduction with Applications, 5th ed., Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03620-4.  Google Scholar  H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995. Google Scholar  P. van den Driessche and X. Zou, Global attractivity in delayed Hopfield neural network models, SIAM J. Appl. Math., 58 (1998), 1878-1890.  doi: 10.1137/S0036139997321219.  Google Scholar  J. Wu, Symmetric functional differential equations and neural networks with memory, Trans. Amer. Math. Soc., 350 (1998), 4799-4838.  doi: 10.1090/S0002-9947-98-02083-2.  Google Scholar
  María J. Garrido–Atienza, Kening Lu, Björn Schmalfuss. Random dynamical systems for stochastic partial differential equations driven by a fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 473-493. doi: 10.3934/dcdsb.2010.14.473  Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745  Robert Hesse, Alexandra Neamţu. Global solutions and random dynamical systems for rough evolution equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2723-2748. doi: 10.3934/dcdsb.2020029  Ludwig Arnold, Igor Chueshov. Cooperative random and stochastic differential equations. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 1-33. doi: 10.3934/dcds.2001.7.1  Neville J. Ford, Stewart J. Norton. Predicting changes in dynamical behaviour in solutions to stochastic delay differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 367-382. doi: 10.3934/cpaa.2006.5.367  Mazyar Ghani Varzaneh, Sebastian Riedel. A dynamical theory for singular stochastic delay differential equations Ⅱ: nonlinear equations and invariant manifolds. Discrete & Continuous Dynamical Systems - B, 2021, 26 (8) : 4587-4612. doi: 10.3934/dcdsb.2020304  Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098  Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157  Kai Liu. Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3915-3934. doi: 10.3934/dcdsb.2018117  Graeme D. Chalmers, Desmond J. Higham. Convergence and stability analysis for implicit simulations of stochastic differential equations with random jump magnitudes. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 47-64. doi: 10.3934/dcdsb.2008.9.47  Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229  Zhong Tan, Leilei Tong. Asymptotic stability of stationary solutions for magnetohydrodynamic equations. Discrete & Continuous Dynamical Systems, 2017, 37 (6) : 3435-3465. doi: 10.3934/dcds.2017146  Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122  Yaofeng Su. Almost surely invariance principle for non-stationary and random intermittent dynamical systems. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6585-6597. doi: 10.3934/dcds.2019286  Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355  Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1  Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521  Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324  Yong Ren, Xuejuan Jia, Lanying Hu. Exponential stability of solutions to impulsive stochastic differential equations driven by $G$-Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2157-2169. doi: 10.3934/dcdsb.2015.20.2157  Wei Mao, Liangjian Hu, Xuerong Mao. Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 587-613. doi: 10.3934/dcdsb.2018198

2019 Impact Factor: 1.27

## Metrics

• PDF downloads (23)
• HTML views (73)
• Cited by (0)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]