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

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 coeffcients. 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 coeffcients 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 ows 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 Duffng 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.