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:
[1]

V. M. AfraĭmovichN. N. Verichev and M. I. Rabinovich, Stochastic synchronization of oscillations in dissipative systems, Izv. Vyssh. Uchebn. Zaved. Radiofiz., 29 (1986), 1050-1060.   Google Scholar

[2]

J. ApplebyG. 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

[3]

J. ApplebyC. KellyX. 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

[4]

J. ApplebyG. 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

[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.  Google Scholar

[6]

J. ApplebyX. 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

[7]

L. Arnold, Stabilization by noise revisited, Z. Angew. Math. Mech., 70 (1990), 235-246.  doi: 10.1002/zamm.19900700704.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[10]

P. Billingsley, Probability and Measure, 3rd ed. , John Wiley & Sons, Inc. , New York, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

B. Doiron, J. Rinzel and A. Reyes, Stochastic synchronization in finite size spiking networks Phys. Rev. E, 74 (2006), 030903, 4pp. Google Scholar

[15]

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

[16]

H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469.   Google Scholar

[17]

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

[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.  Google Scholar

[19]

D. J. HighamX. 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

[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.  Google Scholar

[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.  Google Scholar

[22] R. A. Horn and C. R. Johnson, Matrix Analysis, 2 ed., Cambridge University Press, Cambridge, 2013.   Google Scholar
[23]

H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math., 131 (1973), 207-248.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[26]

C. Laing and G. J. Lord (eds. ), Stochastic Methods in Neuroscience, Oxford University Press, Oxford, 2010. Google Scholar

[27]

A. Longtin, Neural coherence and stochastic resonance, in Stochastic Methods in Neuroscience, Oxford Univ. Press, Oxford, 2010, 94-123. Google Scholar

[28]

X. Mao, Stochastic stabilization and destabilization, Systems Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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:
[1]

V. M. AfraĭmovichN. N. Verichev and M. I. Rabinovich, Stochastic synchronization of oscillations in dissipative systems, Izv. Vyssh. Uchebn. Zaved. Radiofiz., 29 (1986), 1050-1060.   Google Scholar

[2]

J. ApplebyG. 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

[3]

J. ApplebyC. KellyX. 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

[4]

J. ApplebyG. 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

[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.  Google Scholar

[6]

J. ApplebyX. 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

[7]

L. Arnold, Stabilization by noise revisited, Z. Angew. Math. Mech., 70 (1990), 235-246.  doi: 10.1002/zamm.19900700704.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[10]

P. Billingsley, Probability and Measure, 3rd ed. , John Wiley & Sons, Inc. , New York, 1995. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

B. Doiron, J. Rinzel and A. Reyes, Stochastic synchronization in finite size spiking networks Phys. Rev. E, 74 (2006), 030903, 4pp. Google Scholar

[15]

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

[16]

H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist., 31 (1960), 457-469.   Google Scholar

[17]

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

[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.  Google Scholar

[19]

D. J. HighamX. 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

[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.  Google Scholar

[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.  Google Scholar

[22] R. A. Horn and C. R. Johnson, Matrix Analysis, 2 ed., Cambridge University Press, Cambridge, 2013.   Google Scholar
[23]

H. Kesten, Random difference equations and renewal theory for products of random matrices, Acta Math., 131 (1973), 207-248.   Google Scholar

[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. Google Scholar

[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.  Google Scholar

[26]

C. Laing and G. J. Lord (eds. ), Stochastic Methods in Neuroscience, Oxford University Press, Oxford, 2010. Google Scholar

[27]

A. Longtin, Neural coherence and stochastic resonance, in Stochastic Methods in Neuroscience, Oxford Univ. Press, Oxford, 2010, 94-123. Google Scholar

[28]

X. Mao, Stochastic stabilization and destabilization, Systems Control Lett., 23 (1994), 279-290.  doi: 10.1016/0167-6911(94)90050-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

Y. Saito and T. Mitsui, Mean-square stability of numerical schemes for stochastic differential systems, Vietnam J. Math., 30 (2002), 551-560.   Google Scholar

Figure 1.  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 $\rho=3$.
[1]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[2]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[3]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[4]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[5]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[6]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[7]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[8]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[9]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054

[10]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[11]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[12]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[13]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[14]

Yangrong Li, Shuang Yang, Qiangheng Zhang. Odd random attractors for stochastic non-autonomous Kuramoto-Sivashinsky equations without dissipation. Electronic Research Archive, 2020, 28 (4) : 1529-1544. doi: 10.3934/era.2020080

[15]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[16]

Ilyasse Lamrani, Imad El Harraki, Ali Boutoulout, Fatima-Zahrae El Alaoui. Feedback stabilization of bilinear coupled hyperbolic systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020434

[17]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

[18]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[19]

Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $ (n, m) $-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117

[20]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (71)
  • HTML views (116)
  • Cited by (1)

Other articles
by authors

[Back to Top]