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An Isaac Newton Institute Workshop

Effective Computational Methods for Highly Oscillatory Problems: The Interplay between Mathematical Theory and Applications

Generalised Polynomial Chaos for Analysing the Stability of Oscillators.

Author: Roland Pulch (University of Wuppertal)


If a dynamical system exhibits a periodic response, analysing the stability of this state yields crucial information. We consider oscillators modelled by systems of ordinary differential equations or differential algebraic equations. Hence we focus on stability properties of periodic solutions with respect to perturbations in corresponding initial values. Floquet theory represents a local concept for analysing the stability. Alternatively, we consider a stochastic perturbation following some probability distribution to obtain global information on stability. This strategy yields a system with stochastic input parameters. Thus results concerning the expected value and the variance of the corresponding solution are of interest. Monte Carlo methods can be used to compute these key figures approximately, where often a huge number of realisations is required. We apply the alternative approach of generalised polynomial chaos to obtain according approximations. Numerical simulations of oscillators using this strategy are presented.