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 & Continuous Dynamical Systems - B, 2017, 22 (2) : 369-381. doi: 10.3934/dcdsb.2017017
##### References:
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Press, Oxford, 2010, 94-123. Google Scholar  X. Mao, Stochastic stabilization and destabilization, Systems Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7. Google Scholar  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. Google Scholar  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. Google Scholar  Y. Saito and T. Mitsui, Mean-square stability of numerical schemes for stochastic differential systems, Vietnam J. Math., 30 (2002), 551-560. Google Scholar

show all references

##### References:
  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  L. Arnold, Stabilization by noise revisited, Z. Angew. Math. Mech., 70 (1990), 235-246.  doi: 10.1002/zamm.19900700704. Google Scholar  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. Google Scholar  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. Google Scholar  P. Billingsley, Probability and Measure, 3rd ed. , John Wiley & Sons, Inc. , New York, 1995. Google Scholar  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. Google Scholar  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. Google Scholar  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. Google Scholar  B. Doiron, J. Rinzel and A. Reyes, Stochastic synchronization in finite size spiking networks Phys. Rev. E, 74 (2006), 030903, 4pp. Google Scholar  M. Freidlin, On stochastic perturbations of dynamical systems with fast and slow components, Stoch. Dyn., 1 (2001), 261-281.  doi: 10.1142/S0219493701000138. Google Scholar  H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469. Google Scholar  D. S. Goldobin and A. Pikovsky, Synchronization and desynchronization of self-sustained oscillators by common noise Phys. Rev. E, 71 (2005), 045201, 4pp. Google Scholar  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. Google Scholar  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. Google Scholar  P. Hitczenko and G. S. Medvedev, Bursting oscillations induced by small noise, SIAM J. Appl. Math., 69 (2009), 1359-1392.  doi: 10.1137/070711803. Google Scholar  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. Google Scholar  R. A. Horn and C. R. Johnson, Matrix Analysis, 2 ed., Cambridge University Press, Cambridge, 2013. Google Scholar  H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math., 131 (1973), 207-248. Google Scholar  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. Google Scholar  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. Google Scholar  C. Laing and G. J. Lord (eds. ), Stochastic Methods in Neuroscience, Oxford University Press, Oxford, 2010. Google Scholar  A. Longtin, Neural coherence and stochastic resonance, in Stochastic Methods in Neuroscience, Oxford Univ. Press, Oxford, 2010, 94-123. Google Scholar  X. Mao, Stochastic stabilization and destabilization, Systems Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7. Google Scholar  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. Google Scholar  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. Google Scholar  Y. Saito and T. Mitsui, Mean-square stability of numerical schemes for stochastic differential systems, Vietnam J. Math., 30 (2002), 551-560. Google Scholar 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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