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Strong averaging principle for slow-fast SPDEs with Poisson random measures
Stochastic averaging for slow-fast dynamical systems with fractional Brownian motion
1. | Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, 710072, China |
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. |
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. |
[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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