Abstract: Many systems of interest in the applied sciences share the common feature of possessing multiple scales, either in time or in space, or both. Some approaches to modelling focus on one scale and incorporate the effect of other scales (e.g. smaller scales) through constitutive relations, which are often obtained empirically. Multiscale modelling approaches are built on the ambition of treating both scales at the same time, with the aim of deriving (rather than empirically obtaining) efficient coarse grained models which incorporate the effects of the smaller/faster scales. In this talk we will consider systems that are multiscale in time; in this context the method of multiscale expansions provides a way to formally derive the CG dynamics, while (stochastic) averaging and homogenization techniques provide analytical tools for rigorous proofs. When using multiscale expansions (as well as averaging and homogenization) a key assumption is that the dynamics for the fast scale is ergodic, i.e. that it has a unique equilibrium (invariant measure). If the fast scale has multiple invariant measures (which is a rather common occurrence in many scenarios) then truly very little is known – multiscale expansions may well not work at all and both averaging and homogenization theory are currently unequipped to help tackle this problem. In this talk we will present situations in which this scenario occurs, recap what is known and point out gaps in current theory.
Based on work with K.Painter and I. Souttar.

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