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September  2015, 20(7): 2257-2267. doi: 10.3934/dcdsb.2015.20.2257

Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion

Yong Xu 1, , Bin Pei 1, and  Rong Guo 1,

1. 

Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710072, China

Received  September 2014 Revised  March 2015 Published  July 2015

This paper investigates the stochastic averaging of slow-fast dynamical systems driven by fractional Brownian motion with the Hurst parameter $H$ in the interval $(\frac{1}{2},1)$. We establish an averaging principle by which the obtained simplified systems (the so-called averaged systems) will be applied to replace the original systems approximately through their solutions. Here, the solutions to averaged equations of slow variables which are unrelated to fast variables can converge to the solutions of slow variables to the original slow-fast dynamical systems in the sense of mean square. Therefore, the dimension reduction is realized since the solutions of uncoupled averaged equations can substitute that of coupled equations of the original slow-fast dynamical systems, namely, the asymptotic solutions dynamics will be obtained by the proposed stochastic averaging approach.
Keywords: averaging, Slow-fast system, mean square, fractional Brownian motion., stochastic differential equations.
Mathematics Subject Classification: Primary: 34F05, 37H10, 60H10, 93E0.
Citation: Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2257-2267. doi: 10.3934/dcdsb.2015.20.2257
References:
[1]

E. Alos and D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stochastics and Stochastic Reports, 75 (2003), 129-152. doi: 10.1080/1045112031000078917.  Google Scholar

[2]

R. Benzi, A. Sutera and A. Vulpiani, The mechanism of stochastic resonance, J. Phys. A, 14 (1981), L453-L457. doi: 10.1088/0305-4470/14/11/006.  Google Scholar

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N. Berglund, B. Gentz and C. Kuehn, Hunting french ducks in a noisy environment, J. Differential Equations, 252 (2012), 4786-4841. doi: 10.1016/j.jde.2012.01.015.  Google Scholar

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N. Chakravarti and K. L. Sebastian, Fractional Brownian motion models for ploymers, Chemical Physics Letter., 267 (1997), 9-13. Google Scholar

[8]

W. Dai and C. C. Heyde, Itô formula with respect to fractional Brownian motion and its application, Journal of Appl. Math. and Stoch. Anal., 9 (1996), 439-448. doi: 10.1155/S104895339600038X.  Google Scholar

[9]

J. Dubbeldam and B. Krauskopf, Self-pulsations in lasers with saturable absorber: Dynamics and bifurcations, Opt. Commun., 159 (1999), 325-338. doi: 10.1016/S0030-4018(98)00568-9.  Google Scholar

[10]

T. Erneux and P. Mandel, Bifurcation phenomena in a laser with a saturable absorber, Z. Phys. B., 44 (1981), 365-374. doi: 10.1007/BF01294174.  Google Scholar

[11]

O. Filatov, Averaging of systems of differential inclusions with slow and fast variables, Differential Equations, 44 (2008), 349-363. doi: 10.1134/S0012266108030063.  Google Scholar

[12]

M. Freidlin and A. Wentzell, Random Perturbations of Dynamical Systems, Springer, 1998. doi: 10.1007/978-1-4612-0611-8.  Google Scholar

[13]

P. Hitczenkoa and G. Medvedev, Bursting oscillations induced by small noise, SIAM J. Appl. Math., 69 (2009), 1359-1392. doi: 10.1137/070711803.  Google Scholar

[14]

Y. Hu and B. Øksendal, Fractional white noise calculus and application to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Topics, 6 (2003), 1-32. doi: 10.1142/S0219025703001110.  Google Scholar

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R. Z. Khasminskii, A limit theorem for the solution of differential equations with random right-hand sides, Theory Probab. Appl., 11 (1966), 390-406. doi: 10.1137/1111038.  Google Scholar

[16]

R. Z. Khasminskii, On the averaging principle for stochastic differential Ito equations, Kybernetika, 4 (1968), 260-279. Google Scholar

[17]

R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Theor. Probab. Appl., 11 (1966), 211-228. Google Scholar

[18]

A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen, Raum, C.R. (Dokaldy) Acad. Sci. URSS (N.S.), 26 (1940), 115-118.  Google Scholar

[19]

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[20]

V. Kolomiets and A. Mel'nikov, Averaging of stochastic systems of integral-differential equations with "Poisson noise", Ukr. Math. J, 43 (1991), 242-246. doi: 10.1007/BF01060515.  Google Scholar

[21]

B. Krauskopf, H. Osinga, J. Galán-Vioque, et al., Mixed-mode Oscillations in a Three Time-Scale Model for the Dopaminergic Neuron, Canopus Publishing Limited, 2007. Google Scholar

[22]

M. Krupa, N. Popovic, N. Kopell and H. Rotstein, Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron, Chaos, 18 (2008), 015106, 19pp. doi: 10.1063/1.2779859.  Google Scholar

[23]

R. Larter, C. Steinmetz and B. Aguda, Fast-slow variable analysis of the transition to mixed-mode oscillations and chaos in the peroxidase reaction, J. Chem. Phys., 89 (1988), 6506-6514. doi: 10.1063/1.455370.  Google Scholar

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W. E. Leland, M. S. Taqqu, W. Willinger and D. V. Wilson, On the self-similar nature of ethernet traffic, IEEE/ACM Trans. Networking., 2 (1994), 1-15. Google Scholar

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R. Liptser and V. Spokoiny, On Estimating a Dynamic Function of a Stochastic System with Averaging, Statistical Inference for Stochastic Processes, 3 (2000), 225-249. doi: 10.1023/A:1009983802178.  Google Scholar

[26]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10 (1968), 422-437. doi: 10.1137/1010093.  Google Scholar

[27]

B. McNamara and K. Wiesenfeld, Theory of stochastic resonance, Physical Review A, 39 (1989), 4854-4869. doi: 10.1103/PhysRevA.39.4854.  Google Scholar

[28]

Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0.  Google Scholar

[29]

N. Sri. Namachchivaya and Y. K. Lin, Application of stochastic averaging for systems with high damping, Probab. Eng. Mech., 3 (1988), 185-196. Google Scholar

[30]

I. Norros, E. Valkeila and J. Virtamo, An elementary approach to a Girsanov formula and other analytivcal resuls on fractional Brownian motion, Bernoulli., 5 (1999), 571-587. doi: 10.2307/3318691.  Google Scholar

[31]

D. Nualart and A. Rascanu, The Malliavin Calculus and Related Topics, Prob. and Appl., 21, Springer-Verlag, 1995. doi: 10.1007/978-1-4757-2437-0.  Google Scholar

[32]

J. Roberts and P. Spanos, Stochastic averaging: An approximate method of solving random vibration problems, Int. J. Non-Linear Mech., 21 (1986), 111-134. doi: 10.1016/0020-7462(86)90025-9.  Google Scholar

[33]

J. Rubin and M. Wechselberger, Giant squid-hidden canard: The 3D geometry of the Hodgkin-Huxley model, Biol. Cyber, 97 (2007), 5-32. doi: 10.1007/s00422-007-0153-5.  Google Scholar

[34]

I. Stoyanov and D. Bainov, The averaging method for a class of stochastic differential equations, Ukr. Math. J., 26 (1974), 186-194. doi: 10.1007/BF01085718.  Google Scholar

[35]

R. L. Stratonovich, Topics in the Theory of Random Noise, Silverman Gordon and Breach Science Publishers, New York-London, 1963.  Google Scholar

[36]

J. Su, J. Rubin and D. Terman, Effects of noise on elliptic bursters, Nonlinearity, 17 (2004), 133-157. doi: 10.1088/0951-7715/17/1/009.  Google Scholar

[37]

J. Swift, P. Hohenberg and G. Ahlers, Stochastic Landau equation with time-dependent drift, Physical Review A., 43 (1991), 6572-6580. doi: 10.1103/PhysRevA.43.6572.  Google Scholar

[38]

M. Torrent and M. San Miguel, Stochastic-dynamics characterization of delayed laser threshold instability with swept control parameter, Physical Review A., 38 (1988), 245-251. doi: 10.1103/PhysRevA.38.245.  Google Scholar

[39]

B. Van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Rev., 1 (1920), 754-762. Google Scholar

[40]

W. Wang and A. Roberts, Average and deviation for slow-fast stochastic partial differential equations, Journal of Differential Equations, 253 (2012), 1265-1286. doi: 10.1016/j.jde.2012.05.011.  Google Scholar

[41]

Y. Xu, J. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Levy noise, Physica D., 240 (2011), 1395-1401. doi: 10.1016/j.physd.2011.06.001.  Google Scholar

[42]

Y. Xu, B. Pei and Y. Li, Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Mathematical Methods in the Applied Sciences., 38 (2015), 2120-2131. doi: 10.1002/mma.3208.  Google Scholar

[43]

Y. Xu, R. Guo, D. Liu, H. Zhang and J. Duan, Stochastic averaging principle for dynamical systems with fractional Brownian motion, Discrete and Continuous Dynamical Systems B, 19 (2014), 1197-1212. doi: 10.3934/dcdsb.2014.19.1197.  Google Scholar

[44]

Y. Xu, B. Pei and Y. Li, An averaging principle for stochastic differential delay equations with fractional Brownian motion, Abstract and Applied Analysis., (2014), Article ID 479195, 10pp. doi: 10.1155/2014/479195.  Google Scholar

show all references

References:
[1]

E. Alos and D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stochastics and Stochastic Reports, 75 (2003), 129-152. doi: 10.1080/1045112031000078917.  Google Scholar

[2]

R. Benzi, A. Sutera and A. Vulpiani, The mechanism of stochastic resonance, J. Phys. A, 14 (1981), L453-L457. doi: 10.1088/0305-4470/14/11/006.  Google Scholar

[3]

N. Berglund and B. Gentz, The effect of additive noise on dynamical hysteresis, Nonlinearity, 15 (2002), 605-632. doi: 10.1088/0951-7715/15/3/305.  Google Scholar

[4]

N. Berglund, B. Gentz and C. Kuehn, Hunting french ducks in a noisy environment, J. Differential Equations, 252 (2012), 4786-4841. doi: 10.1016/j.jde.2012.01.015.  Google Scholar

[5]

F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84628-797-8.  Google Scholar

[6]

P. Braza and T. Erneux, Singular Hopf bifurcation to unstable periodic solutions in an NMR laser, Physical Review A, 40 (1989), p2539. doi: 10.1103/PhysRevA.40.2539.  Google Scholar

[7]

N. Chakravarti and K. L. Sebastian, Fractional Brownian motion models for ploymers, Chemical Physics Letter., 267 (1997), 9-13. Google Scholar

[8]

W. Dai and C. C. Heyde, Itô formula with respect to fractional Brownian motion and its application, Journal of Appl. Math. and Stoch. Anal., 9 (1996), 439-448. doi: 10.1155/S104895339600038X.  Google Scholar

[9]

J. Dubbeldam and B. Krauskopf, Self-pulsations in lasers with saturable absorber: Dynamics and bifurcations, Opt. Commun., 159 (1999), 325-338. doi: 10.1016/S0030-4018(98)00568-9.  Google Scholar

[10]

T. Erneux and P. Mandel, Bifurcation phenomena in a laser with a saturable absorber, Z. Phys. B., 44 (1981), 365-374. doi: 10.1007/BF01294174.  Google Scholar

[11]

O. Filatov, Averaging of systems of differential inclusions with slow and fast variables, Differential Equations, 44 (2008), 349-363. doi: 10.1134/S0012266108030063.  Google Scholar

[12]

M. Freidlin and A. Wentzell, Random Perturbations of Dynamical Systems, Springer, 1998. doi: 10.1007/978-1-4612-0611-8.  Google Scholar

[13]

P. Hitczenkoa and G. Medvedev, Bursting oscillations induced by small noise, SIAM J. Appl. Math., 69 (2009), 1359-1392. doi: 10.1137/070711803.  Google Scholar

[14]

Y. Hu and B. Øksendal, Fractional white noise calculus and application to finance, Infin. Dimens. Anal. Quantum Probab. Relat. Topics, 6 (2003), 1-32. doi: 10.1142/S0219025703001110.  Google Scholar

[15]

R. Z. Khasminskii, A limit theorem for the solution of differential equations with random right-hand sides, Theory Probab. Appl., 11 (1966), 390-406. doi: 10.1137/1111038.  Google Scholar

[16]

R. Z. Khasminskii, On the averaging principle for stochastic differential Ito equations, Kybernetika, 4 (1968), 260-279. Google Scholar

[17]

R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Theor. Probab. Appl., 11 (1966), 211-228. Google Scholar

[18]

A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen, Raum, C.R. (Dokaldy) Acad. Sci. URSS (N.S.), 26 (1940), 115-118.  Google Scholar

[19]

M. Koper, Bifurcations of mixed-mode oscillations in a threevariable autonomous Van der Pol-Duffing model with a cross-shaped phase diagram, Physica D, 80 (1995), 72-94. doi: 10.1016/0167-2789(95)90061-6.  Google Scholar

[20]

V. Kolomiets and A. Mel'nikov, Averaging of stochastic systems of integral-differential equations with "Poisson noise", Ukr. Math. J, 43 (1991), 242-246. doi: 10.1007/BF01060515.  Google Scholar

[21]

B. Krauskopf, H. Osinga, J. Galán-Vioque, et al., Mixed-mode Oscillations in a Three Time-Scale Model for the Dopaminergic Neuron, Canopus Publishing Limited, 2007. Google Scholar

[22]

M. Krupa, N. Popovic, N. Kopell and H. Rotstein, Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron, Chaos, 18 (2008), 015106, 19pp. doi: 10.1063/1.2779859.  Google Scholar

[23]

R. Larter, C. Steinmetz and B. Aguda, Fast-slow variable analysis of the transition to mixed-mode oscillations and chaos in the peroxidase reaction, J. Chem. Phys., 89 (1988), 6506-6514. doi: 10.1063/1.455370.  Google Scholar

[24]

W. E. Leland, M. S. Taqqu, W. Willinger and D. V. Wilson, On the self-similar nature of ethernet traffic, IEEE/ACM Trans. Networking., 2 (1994), 1-15. Google Scholar

[25]

R. Liptser and V. Spokoiny, On Estimating a Dynamic Function of a Stochastic System with Averaging, Statistical Inference for Stochastic Processes, 3 (2000), 225-249. doi: 10.1023/A:1009983802178.  Google Scholar

[26]

B. B. Mandelbrot and J. W. Van Ness, Fractional Brownian motions, fractional noises and applications, SIAM Review, 10 (1968), 422-437. doi: 10.1137/1010093.  Google Scholar

[27]

B. McNamara and K. Wiesenfeld, Theory of stochastic resonance, Physical Review A, 39 (1989), 4854-4869. doi: 10.1103/PhysRevA.39.4854.  Google Scholar

[28]

Y. S. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-75873-0.  Google Scholar

[29]

N. Sri. Namachchivaya and Y. K. Lin, Application of stochastic averaging for systems with high damping, Probab. Eng. Mech., 3 (1988), 185-196. Google Scholar

[30]

I. Norros, E. Valkeila and J. Virtamo, An elementary approach to a Girsanov formula and other analytivcal resuls on fractional Brownian motion, Bernoulli., 5 (1999), 571-587. doi: 10.2307/3318691.  Google Scholar

[31]

D. Nualart and A. Rascanu, The Malliavin Calculus and Related Topics, Prob. and Appl., 21, Springer-Verlag, 1995. doi: 10.1007/978-1-4757-2437-0.  Google Scholar

[32]

J. Roberts and P. Spanos, Stochastic averaging: An approximate method of solving random vibration problems, Int. J. Non-Linear Mech., 21 (1986), 111-134. doi: 10.1016/0020-7462(86)90025-9.  Google Scholar

[33]

J. Rubin and M. Wechselberger, Giant squid-hidden canard: The 3D geometry of the Hodgkin-Huxley model, Biol. Cyber, 97 (2007), 5-32. doi: 10.1007/s00422-007-0153-5.  Google Scholar

[34]

I. Stoyanov and D. Bainov, The averaging method for a class of stochastic differential equations, Ukr. Math. J., 26 (1974), 186-194. doi: 10.1007/BF01085718.  Google Scholar

[35]

R. L. Stratonovich, Topics in the Theory of Random Noise, Silverman Gordon and Breach Science Publishers, New York-London, 1963.  Google Scholar

[36]

J. Su, J. Rubin and D. Terman, Effects of noise on elliptic bursters, Nonlinearity, 17 (2004), 133-157. doi: 10.1088/0951-7715/17/1/009.  Google Scholar

[37]

J. Swift, P. Hohenberg and G. Ahlers, Stochastic Landau equation with time-dependent drift, Physical Review A., 43 (1991), 6572-6580. doi: 10.1103/PhysRevA.43.6572.  Google Scholar

[38]

M. Torrent and M. San Miguel, Stochastic-dynamics characterization of delayed laser threshold instability with swept control parameter, Physical Review A., 38 (1988), 245-251. doi: 10.1103/PhysRevA.38.245.  Google Scholar

[39]

B. Van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Rev., 1 (1920), 754-762. Google Scholar

[40]

W. Wang and A. Roberts, Average and deviation for slow-fast stochastic partial differential equations, Journal of Differential Equations, 253 (2012), 1265-1286. doi: 10.1016/j.jde.2012.05.011.  Google Scholar

[41]

Y. Xu, J. Duan and W. Xu, An averaging principle for stochastic dynamical systems with Levy noise, Physica D., 240 (2011), 1395-1401. doi: 10.1016/j.physd.2011.06.001.  Google Scholar

[42]

Y. Xu, B. Pei and Y. Li, Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise, Mathematical Methods in the Applied Sciences., 38 (2015), 2120-2131. doi: 10.1002/mma.3208.  Google Scholar

[43]

Y. Xu, R. Guo, D. Liu, H. Zhang and J. Duan, Stochastic averaging principle for dynamical systems with fractional Brownian motion, Discrete and Continuous Dynamical Systems B, 19 (2014), 1197-1212. doi: 10.3934/dcdsb.2014.19.1197.  Google Scholar

[44]

Y. Xu, B. Pei and Y. Li, An averaging principle for stochastic differential delay equations with fractional Brownian motion, Abstract and Applied Analysis., (2014), Article ID 479195, 10pp. doi: 10.1155/2014/479195.  Google Scholar

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