Polynomial chaos based uncertainty quantification in Hamiltonian and chaotic systems

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 coefficients. Here we study the structure and dynamics of these differential equations when the original system has Hamiltonian structure or displays chaotic dynamics. In particular, we prove that the differential equations for the expansion coefficients in generalized polynomial chaos expansions of Hamiltonian systems retain the Hamiltonian structure relative to the ensemble average Hamiltonian. Additionally, 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.