# American Institute of Mathematical Sciences

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

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

 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.
Citation: Yong Xu, Bin Pei, Rong Guo. Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion. Discrete and 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. [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. [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. [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. [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. [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. [7] N. Chakravarti and K. L. Sebastian, Fractional Brownian motion models for ploymers, Chemical Physics Letter., 267 (1997), 9-13. [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. [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. [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. [11] O. Filatov, Averaging of systems of differential inclusions with slow and fast variables, Differential Equations, 44 (2008), 349-363. doi: 10.1134/S0012266108030063. [12] M. Freidlin and A. Wentzell, Random Perturbations of Dynamical Systems, Springer, 1998. doi: 10.1007/978-1-4612-0611-8. [13] P. Hitczenkoa and G. Medvedev, Bursting oscillations induced by small noise, SIAM J. Appl. Math., 69 (2009), 1359-1392. doi: 10.1137/070711803. [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. [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. [16] R. Z. Khasminskii, On the averaging principle for stochastic differential Ito equations, Kybernetika, 4 (1968), 260-279. [17] R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Theor. Probab. Appl., 11 (1966), 211-228. [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. [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. [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. [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. [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. [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. [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. [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. [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. [27] B. McNamara and K. Wiesenfeld, Theory of stochastic resonance, Physical Review A, 39 (1989), 4854-4869. doi: 10.1103/PhysRevA.39.4854. [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. [29] N. Sri. Namachchivaya and Y. K. Lin, Application of stochastic averaging for systems with high damping, Probab. Eng. Mech., 3 (1988), 185-196. [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. [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. [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. [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. [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. [35] R. L. Stratonovich, Topics in the Theory of Random Noise, Silverman Gordon and Breach Science Publishers, New York-London, 1963. [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. [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. [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. [39] B. Van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Rev., 1 (1920), 754-762. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [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. [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. [7] N. Chakravarti and K. L. Sebastian, Fractional Brownian motion models for ploymers, Chemical Physics Letter., 267 (1997), 9-13. [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. [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. [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. [11] O. Filatov, Averaging of systems of differential inclusions with slow and fast variables, Differential Equations, 44 (2008), 349-363. doi: 10.1134/S0012266108030063. [12] M. Freidlin and A. Wentzell, Random Perturbations of Dynamical Systems, Springer, 1998. doi: 10.1007/978-1-4612-0611-8. [13] P. Hitczenkoa and G. Medvedev, Bursting oscillations induced by small noise, SIAM J. Appl. Math., 69 (2009), 1359-1392. doi: 10.1137/070711803. [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. [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. [16] R. Z. Khasminskii, On the averaging principle for stochastic differential Ito equations, Kybernetika, 4 (1968), 260-279. [17] R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Theor. Probab. Appl., 11 (1966), 211-228. [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. [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. [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. [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. [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. [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. [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. [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. [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. [27] B. McNamara and K. Wiesenfeld, Theory of stochastic resonance, Physical Review A, 39 (1989), 4854-4869. doi: 10.1103/PhysRevA.39.4854. [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. [29] N. Sri. Namachchivaya and Y. K. Lin, Application of stochastic averaging for systems with high damping, Probab. Eng. Mech., 3 (1988), 185-196. [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. [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. [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. [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. [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. [35] R. L. Stratonovich, Topics in the Theory of Random Noise, Silverman Gordon and Breach Science Publishers, New York-London, 1963. [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. [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. [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. [39] B. Van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Rev., 1 (1920), 754-762. [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. [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. [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. [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. [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.
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