# American Institute of Mathematical Sciences

March  2017, 22(2): 369-381. doi: 10.3934/dcdsb.2017017

## Stability of equilibria of randomly perturbed maps

 Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA

* Corresponding author: Georgi S. Medvedev

Received  March 2016 Revised  September 2016 Published  December 2016

Fund Project: PH was supported by a grant from Simons Foundation grant #208766. GSM was supported by the NSF grant DMS #1412066. GSM participated in a SQuaRe group 'Stochastic stabilisation of limit-cycle dynamics in ecology and neuroscience' sponsored by the American Institute of Mathematics.

We derive a sufficient condition for stability in probability of an equilibrium of a randomly perturbed map in $\mathbb{R}^d$. This condition can be used to stabilize unstable equilibria by random forcing. Analytical results on stabilization are illustrated with numerical examples of randomly perturbed nonlinear maps in one-and two-dimensional spaces.

Citation: PaweŁ Hitczenko, Georgi S. Medvedev. Stability of equilibria of randomly perturbed maps. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 369-381. doi: 10.3934/dcdsb.2017017
##### References:
 [1] V. M. Afraĭmovich, N. N. Verichev and M. I. Rabinovich, Stochastic synchronization of oscillations in dissipative systems, Izv. Vyssh. Uchebn. Zaved. Radiofiz., 29 (1986), 1050-1060. [2] J. Appleby, G. Berkolaiko and A. Rodkina, On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations, J. Difference Equ. Appl., 14 (2008), 923-951.  doi: 10.1080/10236190701871786. [3] J. Appleby, C. Kelly, X. Mao and A. Rodkina, On the local dynamics of polynomial difference equations with fading stochastic perturbations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17 (2010), 401-430. [4] J. Appleby, G. Berkolaiko and A. Rodkina, Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise, Stochastics, 81 (2009), 99-127.  doi: 10.1080/17442500802088541. [5] J. Appleby and X. Mao, Stochastic stabilisation of functional differential equations, Systems Control Lett., 54 (2005), 1069-1081.  doi: 10.1016/j.sysconle.2005.03.003. [6] J. Appleby, X. Mao and A. Rodkina, On stochastic stabilization of difference equations, Discrete Contin. Dyn. Syst., 15 (2006), 843-857.  doi: 10.3934/dcds.2006.15.843. [7] L. Arnold, Stabilization by noise revisited, Z. Angew. Math. Mech., 70 (1990), 235-246.  doi: 10.1002/zamm.19900700704. [8] N. Berglund and B. Gentz, Noise-induced Phenomena in Slow-Fast Dynamical Systems, Probability and its Applications (New York), Springer-Verlag London, Ltd. , London, 2006. [9] G. Berkolaiko and A. Rodkina, Almost sure convergence of solutions to nonhomogeneous stochastic difference equation, J. Difference Equ. Appl., 12 (2006), 535-553.  doi: 10.1080/10236190600574093. [10] P. Billingsley, Probability and Measure, 3rd ed. , John Wiley & Sons, Inc. , New York, 1995. [11] E. Braverman and A. Rodkina, On difference equations with asymptotically stable 2-cycles perturbed by a decaying noise, Comput. Math. Appl., 64 (2012), 2224-2232.  doi: 10.1016/j.camwa.2012.01.057. [12] E. Buckwar and C. Kelly, Towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), 298-321.  doi: 10.1137/090771843. [13] R. E. L. DeVille, E. Vanden-Eijnden and C. B. Muratov, Two distinct mechanisms of coherence in randomly perturbed dynamical system Phys. Rev. E, 72 (2005), 031105, 10pp. [14] B. Doiron, J. Rinzel and A. Reyes, Stochastic synchronization in finite size spiking networks Phys. Rev. E, 74 (2006), 030903, 4pp. [15] M. Freidlin, On stochastic perturbations of dynamical systems with fast and slow components, Stoch. Dyn., 1 (2001), 261-281.  doi: 10.1142/S0219493701000138. [16] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469. [17] D. S. Goldobin and A. Pikovsky, Synchronization and desynchronization of self-sustained oscillators by common noise Phys. Rev. E, 71 (2005), 045201, 4pp. [18] D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38 (2000), 753-769 (electronic).  doi: 10.1137/S003614299834736X. [19] D. J. Higham, X. Mao and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal., 45 (2007), 592-609 (electronic).  doi: 10.1137/060658138. [20] P. Hitczenko and G. S. Medvedev, Bursting oscillations induced by small noise, SIAM J. Appl. Math., 69 (2009), 1359-1392.  doi: 10.1137/070711803. [21] P. Hitczenko and G. S. Medvedev, The Poincaré map of randomly perturbed periodic motion, J. Nonlinear Sci., 23 (2013), 835-861.  doi: 10.1007/s00332-013-9170-9. [22] R. A. Horn and C. R. Johnson, Matrix Analysis, 2 ed., Cambridge University Press, Cambridge, 2013. [23] H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math., 131 (1973), 207-248. [24] R. Khasminskii, Stochastic Stability of Differential Equations With contributions by G. N. Milstein and M. B. Nevelson, second ed. , Stochastic Modelling and Applied Probability, vol. 66, Springer, Heidelberg, 2012. [25] H. Koçak and K. J. Palmer, Lyapunov exponents and sensitivity dependence, J. Dynam. Differential Equations, 22 (2010), 381-398.  doi: 10.1007/s10884-010-9169-y. [26] C. Laing and G. J. Lord (eds. ), Stochastic Methods in Neuroscience, Oxford University Press, Oxford, 2010. [27] A. Longtin, Neural coherence and stochastic resonance, in Stochastic Methods in Neuroscience, Oxford Univ. Press, Oxford, 2010, 94-123. [28] X. Mao, Stochastic stabilization and destabilization, Systems Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7. [29] M. Porfiri and R. Pigliacampo, Master-slave global stochastic synchronization of chaotic oscillators, SIAM J. Appl. Dyn. Syst., 7 (2008), 825-842.  doi: 10.1137/070688973. [30] Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267.  doi: 10.1137/S0036142992228409. [31] Y. Saito and T. Mitsui, Mean-square stability of numerical schemes for stochastic differential systems, Vietnam J. Math., 30 (2002), 551-560.

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##### References:
 [1] V. M. Afraĭmovich, N. N. Verichev and M. I. Rabinovich, Stochastic synchronization of oscillations in dissipative systems, Izv. Vyssh. Uchebn. Zaved. Radiofiz., 29 (1986), 1050-1060. [2] J. Appleby, G. Berkolaiko and A. Rodkina, On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations, J. Difference Equ. Appl., 14 (2008), 923-951.  doi: 10.1080/10236190701871786. [3] J. Appleby, C. Kelly, X. Mao and A. Rodkina, On the local dynamics of polynomial difference equations with fading stochastic perturbations, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17 (2010), 401-430. [4] J. Appleby, G. Berkolaiko and A. Rodkina, Non-exponential stability and decay rates in nonlinear stochastic difference equations with unbounded noise, Stochastics, 81 (2009), 99-127.  doi: 10.1080/17442500802088541. [5] J. Appleby and X. Mao, Stochastic stabilisation of functional differential equations, Systems Control Lett., 54 (2005), 1069-1081.  doi: 10.1016/j.sysconle.2005.03.003. [6] J. Appleby, X. Mao and A. Rodkina, On stochastic stabilization of difference equations, Discrete Contin. Dyn. Syst., 15 (2006), 843-857.  doi: 10.3934/dcds.2006.15.843. [7] L. Arnold, Stabilization by noise revisited, Z. Angew. Math. Mech., 70 (1990), 235-246.  doi: 10.1002/zamm.19900700704. [8] N. Berglund and B. Gentz, Noise-induced Phenomena in Slow-Fast Dynamical Systems, Probability and its Applications (New York), Springer-Verlag London, Ltd. , London, 2006. [9] G. Berkolaiko and A. Rodkina, Almost sure convergence of solutions to nonhomogeneous stochastic difference equation, J. Difference Equ. Appl., 12 (2006), 535-553.  doi: 10.1080/10236190600574093. [10] P. Billingsley, Probability and Measure, 3rd ed. , John Wiley & Sons, Inc. , New York, 1995. [11] E. Braverman and A. Rodkina, On difference equations with asymptotically stable 2-cycles perturbed by a decaying noise, Comput. Math. Appl., 64 (2012), 2224-2232.  doi: 10.1016/j.camwa.2012.01.057. [12] E. Buckwar and C. Kelly, Towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations, SIAM J. Numer. Anal., 48 (2010), 298-321.  doi: 10.1137/090771843. [13] R. E. L. DeVille, E. Vanden-Eijnden and C. B. Muratov, Two distinct mechanisms of coherence in randomly perturbed dynamical system Phys. Rev. E, 72 (2005), 031105, 10pp. [14] B. Doiron, J. Rinzel and A. Reyes, Stochastic synchronization in finite size spiking networks Phys. Rev. E, 74 (2006), 030903, 4pp. [15] M. Freidlin, On stochastic perturbations of dynamical systems with fast and slow components, Stoch. Dyn., 1 (2001), 261-281.  doi: 10.1142/S0219493701000138. [16] H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469. [17] D. S. Goldobin and A. Pikovsky, Synchronization and desynchronization of self-sustained oscillators by common noise Phys. Rev. E, 71 (2005), 045201, 4pp. [18] D. J. Higham, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38 (2000), 753-769 (electronic).  doi: 10.1137/S003614299834736X. [19] D. J. Higham, X. Mao and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal., 45 (2007), 592-609 (electronic).  doi: 10.1137/060658138. [20] P. Hitczenko and G. S. Medvedev, Bursting oscillations induced by small noise, SIAM J. Appl. Math., 69 (2009), 1359-1392.  doi: 10.1137/070711803. [21] P. Hitczenko and G. S. Medvedev, The Poincaré map of randomly perturbed periodic motion, J. Nonlinear Sci., 23 (2013), 835-861.  doi: 10.1007/s00332-013-9170-9. [22] R. A. Horn and C. R. Johnson, Matrix Analysis, 2 ed., Cambridge University Press, Cambridge, 2013. [23] H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math., 131 (1973), 207-248. [24] R. Khasminskii, Stochastic Stability of Differential Equations With contributions by G. N. Milstein and M. B. Nevelson, second ed. , Stochastic Modelling and Applied Probability, vol. 66, Springer, Heidelberg, 2012. [25] H. Koçak and K. J. Palmer, Lyapunov exponents and sensitivity dependence, J. Dynam. Differential Equations, 22 (2010), 381-398.  doi: 10.1007/s10884-010-9169-y. [26] C. Laing and G. J. Lord (eds. ), Stochastic Methods in Neuroscience, Oxford University Press, Oxford, 2010. [27] A. Longtin, Neural coherence and stochastic resonance, in Stochastic Methods in Neuroscience, Oxford Univ. Press, Oxford, 2010, 94-123. [28] X. Mao, Stochastic stabilization and destabilization, Systems Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7. [29] M. Porfiri and R. Pigliacampo, Master-slave global stochastic synchronization of chaotic oscillators, SIAM J. Appl. Dyn. Syst., 7 (2008), 825-842.  doi: 10.1137/070688973. [30] Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267.  doi: 10.1137/S0036142992228409. [31] Y. Saito and T. Mitsui, Mean-square stability of numerical schemes for stochastic differential systems, Vietnam J. Math., 30 (2002), 551-560.
a) Trajectories of the logistic map $x\mapsto f(x)$ (plotted in dashed line) and that of the randomly perturbed system (1). The former approaches the stable equilibrium of the deterministic system $\bar x_2$, while the latter returns to and remains in a small neighborhood of the origin. b) A trajectory of the two-dimesional system (46) stays near the orgin after brief transients. All trajectories of the underlying deterministic system $x\mapsto Ax+q(x)$ starting off the $x^{(2)}$-axis tend to infinity. In numerical simulations shown in a and b, the following parameter values were used: $\epsilon=0.05$ and $\rho=3$.
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