Exponentially accurate balance dynamics

Abstract

By explicitly bounding the growth of terms in a singular perturbation expansion with a small parameter ${\varepsilon}$, we show that it is possible to find a solution that satisfies a balance relation (which defines the slow manifold) up to an error that scales exponentially in ${\varepsilon}$ as ${\varepsilon}\to0$. This is first done for a generic finite-dimensional dynamical system with polynomial nonlinearity, followed by a continuous fluid case. In addition, for the finite-dimensional system, we show that, properly initialized, the solution of the full model stays within an exponential distance to that of the balance equation (i.e., evolution on the slow manifold) over a timescale of order one (independent of ${\varepsilon}$).