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Analysis of a nonlocal-in-time parabolic equation
Stability of equilibria of randomly perturbed maps
Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA |
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.
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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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. 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. |
[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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[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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