Averaging method for differential equations perturbed by dynamical systems

Françoise Pène

ESAIM: Probability and Statistics (2010)

  • Volume: 6, page 33-88
  • ISSN: 1292-8100

Abstract

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In this paper, we are interested in the asymptotical behavior of the error between the solution of a differential equation perturbed by a flow (or by a transformation) and the solution of the associated averaged differential equation. The main part of this redaction is devoted to the ascertainment of results of convergence in distribution analogous to those obtained in [10] and [11]. As in [11], we shall use a representation by a suspension flow over a dynamical system. Here, we make an assumption of multiple decorrelation in terms of this dynamical system. We show how this property can be verified for ergodic algebraic toral automorphisms and point out the fact that, for two-dimensional dispersive billiards, it is a consequence of the method developed in [18]. Moreover, the singular case of a degenerated limit distribution is also considered.

How to cite

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Pène, Françoise. "Averaging method for differential equations perturbed by dynamical systems." ESAIM: Probability and Statistics 6 (2010): 33-88. <http://eudml.org/doc/104296>.

@article{Pène2010,
abstract = { In this paper, we are interested in the asymptotical behavior of the error between the solution of a differential equation perturbed by a flow (or by a transformation) and the solution of the associated averaged differential equation. The main part of this redaction is devoted to the ascertainment of results of convergence in distribution analogous to those obtained in [10] and [11]. As in [11], we shall use a representation by a suspension flow over a dynamical system. Here, we make an assumption of multiple decorrelation in terms of this dynamical system. We show how this property can be verified for ergodic algebraic toral automorphisms and point out the fact that, for two-dimensional dispersive billiards, it is a consequence of the method developed in [18]. Moreover, the singular case of a degenerated limit distribution is also considered. },
author = {Pène, Françoise},
journal = {ESAIM: Probability and Statistics},
keywords = {Dynamical system; hyperbolicity; billiard; suspension flow; limit theorem; averaging method; perturbation; differential equation.; exponential decay of correlations; limit theorem; toral automorphism; Sinai billiards},
language = {eng},
month = {3},
pages = {33-88},
publisher = {EDP Sciences},
title = {Averaging method for differential equations perturbed by dynamical systems},
url = {http://eudml.org/doc/104296},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Pène, Françoise
TI - Averaging method for differential equations perturbed by dynamical systems
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 33
EP - 88
AB - In this paper, we are interested in the asymptotical behavior of the error between the solution of a differential equation perturbed by a flow (or by a transformation) and the solution of the associated averaged differential equation. The main part of this redaction is devoted to the ascertainment of results of convergence in distribution analogous to those obtained in [10] and [11]. As in [11], we shall use a representation by a suspension flow over a dynamical system. Here, we make an assumption of multiple decorrelation in terms of this dynamical system. We show how this property can be verified for ergodic algebraic toral automorphisms and point out the fact that, for two-dimensional dispersive billiards, it is a consequence of the method developed in [18]. Moreover, the singular case of a degenerated limit distribution is also considered.
LA - eng
KW - Dynamical system; hyperbolicity; billiard; suspension flow; limit theorem; averaging method; perturbation; differential equation.; exponential decay of correlations; limit theorem; toral automorphism; Sinai billiards
UR - http://eudml.org/doc/104296
ER -

References

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  14. F. Pène, Applications des propriétés stochastiques des systèmes dynamiques de type hyperbolique : ergodicité du billard dispersif dans le plan, moyennisation d'équations différentielles perturbées par un flot ergodique, Ph.D. Thesis. University of Rennes I, France (2000).  
  15. F. Pène, Rates of convergence in the CLT for two-dimensional dispersive billiards. Comm. Math. Phys.225 (2002) 91-119.  Zbl1040.37023
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