Edgeworth expansions for slow-fast systems with finite time scale separation. (arXiv:1708.06984v2 [nlin.CD] UPDATED)

We derive Edgeworth expansions that describe corrections to the Gaussian limiting behaviour of deterministic slow-fast systems. The Edgeworth expansion is achieved using a semi-group formalism for the transfer operator, where a Duhamel-Dyson series is used to asymptotically determine the corrections at any desired order of the time scale parameter $\varepsilon$. The Edgeworth corrections describe deviations from Gaussianity due to the finite time scale separation and involve integrals over higher-order auto-correlation functions. We develop a diagrammatic representation of the series to control the combinatorial wealth of the asymptotic expansion in $\varepsilon$ and provide explicit expressions for the first two orders. Our method provides an improvement on the classical homogenization limit which is restricted to the limit of infinite time scale separation. We corroborate our analytical results with numerical simulations. 查看全文>>