# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021133
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## 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 Early access 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:
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##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [2] I. Chueshov, Monotone Random Systems Theory and Applications, Lecture Notes in Mathematics, vol. 1779, Springer-Verlag, Berlin, 2002. doi: 10.1007/b83277.  Google Scholar [3] 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 [4] 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 [5] 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 [6] 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 [7] R. Kha'sminskii, Stochastic Stability of Differential Equations, Alphen: Sijtjoff and Noordhoff, 1980. Google Scholar [8] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, Cambridge, 1990.   Google Scholar [9] 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 [10] X. Mao, Stochastic Differential Equations and Applications, Horwood, Chichester, 1997.  Google Scholar [11] X. Mao, Stochastic versions of the LaSalle theorem, J. Differential Equations, 153 (1999), 175-195.  doi: 10.1006/jdeq.1998.3552.  Google Scholar [12] 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 [13] 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 [14] 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 [15] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, 1995.  Google Scholar [16] 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 [17] 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
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