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

*(English)*Zbl 1335.34090Summary: 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.

##### MSC:

34F05 | Ordinary differential equations and systems with randomness |

37H10 | Generation, random and stochastic difference and differential equations |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

34E15 | Singular perturbations, general theory for ordinary differential equations |

34C29 | Averaging method for ordinary differential equations |

##### Keywords:

slow-fast system; averaging; stochastic differential equations; mean square; fractional Brownian motion
PDF
BibTeX
XML
Cite

\textit{Y. Xu} et al., Discrete Contin. Dyn. Syst., Ser. B 20, No. 7, 2257--2267 (2015; Zbl 1335.34090)

Full Text:
DOI

##### References:

[1] | E. Alos, Stochastic integration with respect to the fractional Brownian motion,, Stochastics and Stochastic Reports, 75, 129, (2003) · Zbl 1028.60048 |

[2] | R. Benzi, The mechanism of stochastic resonance,, J. Phys. A, 14, (1981) |

[3] | N. Berglund, The effect of additive noise on dynamical hysteresis,, Nonlinearity, 15, 605, (2002) · Zbl 1073.37061 |

[4] | N. Berglund, Hunting french ducks in a noisy environment,, J. Differential Equations, 252, 4786, (2012) · Zbl 1293.37025 |

[5] | F. Biagini, <em>Stochastic Calculus for Fractional Brownian Motion and Applications</em>,, Springer-Verlag, (2008) · Zbl 1157.60002 |

[6] | P. Braza, Singular Hopf bifurcation to unstable periodic solutions in an NMR laser,, Physical Review A, 40, (1989) |

[7] | N. Chakravarti, Fractional Brownian motion models for ploymers,, Chemical Physics Letter., 267, 9, (1997) |

[8] | W. Dai, Itô formula with respect to fractional Brownian motion and its application,, Journal of Appl. Math. and Stoch. Anal., 9, 439, (1996) · Zbl 0867.60029 |

[9] | J. Dubbeldam, Self-pulsations in lasers with saturable absorber: Dynamics and bifurcations,, Opt. Commun., 159, 325, (1999) |

[10] | T. Erneux, Bifurcation phenomena in a laser with a saturable absorber,, Z. Phys. B., 44, 365, (1981) |

[11] | O. Filatov, Averaging of systems of differential inclusions with slow and fast variables,, Differential Equations, 44, 349, (2008) · Zbl 1162.34008 |

[12] | M. Freidlin, <em>Random Perturbations of Dynamical Systems</em>,, Springer, (1998) · Zbl 0922.60006 |

[13] | P. Hitczenkoa, Bursting oscillations induced by small noise,, SIAM J. Appl. Math., 69, 1359, (2009) · Zbl 1176.60044 |

[14] | Y. Hu, Fractional white noise calculus and application to finance,, Infin. Dimens. Anal. Quantum Probab. Relat. Topics, 6, 1, (2003) · Zbl 1045.60072 |

[15] | R. Z. Khasminskii, A limit theorem for the solution of differential equations with random right-hand sides,, Theory Probab. Appl., 11, 390, (1966) |

[16] | R. Z. Khasminskii, On the averaging principle for stochastic differential Ito equations,, Kybernetika, 4, 260, (1968) |

[17] | R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter,, Theor. Probab. Appl., 11, 211, (1966) · Zbl 0168.16002 |

[18] | A. N. Kolmogorov, Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen,, Raum, 26, 115, (1940) · JFM 66.0552.03 |

[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, 72, (1995) · Zbl 0889.34034 |

[20] | V. Kolomiets, Averaging of stochastic systems of integral-differential equations with “Poisson noise”,, Ukr. Math. J, 43, 242, (1991) · Zbl 0735.60060 |

[21] | B. Krauskopf, <em>Mixed-mode Oscillations in a Three Time-Scale Model for the Dopaminergic Neuron</em>,, Canopus Publishing Limited, (2007) |

[22] | M. Krupa, Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron,, Chaos, 18, (2008) · Zbl 1306.34057 |

[23] | R. Larter, Fast-slow variable analysis of the transition to mixed-mode oscillations and chaos in the peroxidase reaction,, J. Chem. Phys., 89, 6506, (1988) |

[24] | W. E. Leland, On the self-similar nature of ethernet traffic,, IEEE/ACM Trans. Networking., 2, 1, (1994) |

[25] | R. Liptser, On Estimating a Dynamic Function of a Stochastic System with Averaging,, Statistical Inference for Stochastic Processes, 3, 225, (2000) · Zbl 0982.62070 |

[26] | B. B. Mandelbrot, Fractional Brownian motions, fractional noises and applications,, SIAM Review, 10, 422, (1968) · Zbl 0179.47801 |

[27] | B. McNamara, Theory of stochastic resonance,, Physical Review A, 39, 4854, (1989) |

[28] | Y. S. Mishura, <em>Stochastic Calculus for Fractional Brownian Motion and Related Processes</em>,, Springer-Verlag, (2008) · Zbl 1138.60006 |

[29] | N. Sri. Namachchivaya, Application of stochastic averaging for systems with high damping,, Probab. Eng. Mech., 3, 185, (1988) · Zbl 0671.93055 |

[30] | I. Norros, An elementary approach to a Girsanov formula and other analytivcal resuls on fractional Brownian motion,, Bernoulli., 5, 571, (1999) · Zbl 0955.60034 |

[31] | D. Nualart, <em>The Malliavin Calculus and Related Topics</em>,, Prob. and Appl., (1995) · Zbl 0837.60050 |

[32] | J. Roberts, Stochastic averaging: An approximate method of solving random vibration problems,, Int. J. Non-Linear Mech., 21, 111, (1986) · Zbl 0582.73077 |

[33] | J. Rubin, Giant squid-hidden canard: The 3D geometry of the Hodgkin-Huxley model,, Biol. Cyber, 97, 5, (2007) · Zbl 1125.92015 |

[34] | I. Stoyanov, The averaging method for a class of stochastic differential equations,, Ukr. Math. J., 26, 186, (1974) · Zbl 0294.60051 |

[35] | R. L. Stratonovich, <em>Topics in the Theory of Random Noise</em>,, Silverman Gordon and Breach Science Publishers, (1963) · Zbl 0119.14502 |

[36] | J. Su, Effects of noise on elliptic bursters,, Nonlinearity, 17, 133, (2004) · Zbl 1083.37533 |

[37] | J. Swift, Stochastic Landau equation with time-dependent drift,, Physical Review A., 43, 6572, (1991) |

[38] | M. Torrent, Stochastic-dynamics characterization of delayed laser threshold instability with swept control parameter,, Physical Review A., 38, 245, (1988) |

[39] | B. Van der Pol, A theory of the amplitude of free and forced triode vibrations,, Radio Rev., 1, 754, (1920) |

[40] | W. Wang, Average and deviation for slow-fast stochastic partial differential equations,, Journal of Differential Equations, 253, 1265, (2012) · Zbl 1251.35201 |

[41] | Y. Xu, An averaging principle for stochastic dynamical systems with Levy noise,, Physica D., 240, 1395, (2011) · Zbl 1236.60060 |

[42] | Y. Xu, Approximation properties for solutions to non-Lipschitz stochastic differential equations with Lévy noise,, Mathematical Methods in the Applied Sciences., 38, 2120, (2015) · Zbl 1345.60051 |

[43] | Y. Xu, Stochastic averaging principle for dynamical systems with fractional Brownian motion,, Discrete and Continuous Dynamical Systems B, 19, 1197, (2014) · Zbl 1314.60122 |

[44] | Y. Xu, An averaging principle for stochastic differential delay equations with fractional Brownian motion,, Abstract and Applied Analysis., (2014) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.