Keywords: probabilistic methods, dynamical systems, stochastic dynamics, infinite dimensional measures.

Nowadays, the analysis of systems involving uncertainty seems to be a major issue in order to ensure their performance and security [1]. Real engineering systems contain uncertainties and variability, on the one hand, the lack of information and the variability of the environment leads to the introduction of parameters which are not precisely known; on the other hand, materials, manufacturing processes and geometry variations introduce supplementary uncertainty.

In this work, we consider dynamic systems containing uncertain parameters and we calculate statistics of some classical elements such as periodic solutions, limit cycles and stability.

The evaluation of such a statistics leads to difficult mathematical problems, since the orbits are elements of an infinite dimensional functional space such as, for instance, the space of Lebesgue quadratic integrable functions defined on a time interval [0, T] and taking their values on an n-dimensional Euclidean space. Thus, it becomes necessary to introduce probability measures in infinite dimensional functional spaces: the usual approach furnished by cylindrical probabilities is not computationally efficient [2] and we introduce a new definition of measures in infinite dimensional spaces, which is one of the originalities of the work.

By using this new theory, the evaluation of statistics of infinitely dimensional random variables may be performed, by using a convenient Hilbertian basis or dense families (such as those furnished by finite element approximations) [3]. Usual regression or projection on polynomial chaos combined with quadrature methods make it possible to compute cumulative distribution functions associated with the representation of an infinitely dimensional random variable in such a basis or family, which leads to a statistical description of the representation [4].

In parallel, a significant effort has been made to study stability problems. Once more, basic concepts have been redefined into a probabilistic way leading to computational methods.

The numerical results show that the method is efficient to calculate statistics and probability distributions. These approaches have been applied with success to analyze the limit cycles of well-known dynamical systems such as mass-damper oscillators and Van der Pol oscillators. Higher order statistics and an accurate confidence interval have been computed, thus making it possible to use this method in optimization, reliability-based optimization or robust control directly.

1

N.V. Sahinidis, "Optimization under uncertainty: state-of-the-art and opportunities", Computers and chemical engineering, 28, 971-983, 2004. doi:10.1016/j.compchemeng.2003.09.017

2

L. Schwartz, "Séminaire d'analyse fonctionnelle (Polytechnique)", 1969, Ann. Math., 48, 385-392, 1947.

3

E. Souza de Cursi, "Representation of solutions in variational calculus", Variational formulations in mechanics: theory and applications, 2007.

4

B. Sudret, M. Berveiller, M. Lemaire, "A stochastic finite element procedure for moment and reliability analysis", European Journal of Computational Mechanics, 8, 825-866, 2006. doi:10.3166/remn.15.825-866

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