  We consider the effect of multiple stochastic parameters on the time-average
quantities of chaotic systems. We employ the recently proposed
\cite{Kantarakias_Papadakis_2023} sensitivity-enhanced generalized polynomial
chaos expansion, se-gPC, to compute efficiently this effect. se-gPC is an
extension of gPC expansion, enriched with the sensitivity of the time-averaged
quantities with respect to the stochastic variables. To compute these
sensitivities, the adjoint of the shadowing operator is derived in the
frequency domain. Coupling the adjoint operator with gPC provides an efficient
uncertainty quantification (UQ) algorithm which, in its simplest form, has
computational cost that is independent of the number of random variables. The
method is applied to the Kuramoto-Sivashinsky equation and is found to produce
results that match very well with Monte-Carlo simulations. The efficiency of
the proposed method significantly outperforms sparse-grid approaches, like
Smolyak Quadrature. These properties make the method suitable for application
to other dynamical systems with many stochastic parameters.
