Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

José Miguel Pasini; Tuhin Sahai

doi:10.3934/jcd.2014.1.357

Article Contents

2014, Volume 1, Issue 2: 357-375. Doi: 10.3934/jcd.2014.1.357

This issue Previous Article Optimizing the stable behavior of parameter-dependent dynamical systems --- maximal domains of attraction, minimal absorption times Next Article Equation-free computation of coarse-grained center manifolds of microscopic simulators

Polynomial chaos based uncertainty quantification in Hamiltonian, multi-time scale, and chaotic systems

José Miguel Pasini^1, and
Tuhin Sahai^1,

1.
United Technologies Research Center, 411 Silver Lane, East Hartford, CT 06108

Received: May 31, 2013

Revised: May 31, 2014

Published: May 2014

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Abstract

Polynomial chaos is a powerful technique for propagating uncertainty through ordinary and partial differential equations. Random variables are expanded in terms of orthogonal polynomials and differential equations are derived for the expansion coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure, multiple time scales, or chaotic dynamics. In particular, we prove that the differential equations for the coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. We connect this with the volume-preserving property of Hamiltonian flows to show that, for an oscillator with uncertain frequency, a finite expansion must fail at long times, regardless of truncation order. Also, using a two-time scale forced nonlinear oscillator, we show that a polynomial chaos expansion of the time-averaged equations captures uncertainty in the slow evolution of the Poincaré section of the system and that, as the scale separation increases, the computational advantage of this procedure increases. Finally, using the forced Duffing oscillator as an example, we demonstrate that when the original dynamical system displays chaotic dynamics, the resulting dynamical system from polynomial chaos also displays chaotic dynamics, limiting its applicability.

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Mathematics Subject Classification: Primary: 34F05, 65P10, 65P20, 34E13, 60H35; Secondary: 37H99.

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