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

Stability of equilibria of randomly perturbed maps

PaweŁ Hitczenko and  Georgi S. Medvedev ,

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.

Keywords: Stochastic difference equation, stochastic stability, stabilization, noise.
Mathematics Subject Classification: Primary: 37H10, 93E15; Secondary: 39A30, 39A50, 34F0.
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. 

show all references

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. 

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 $\rho=3$.
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