Analysis of reduction methods for multiscale problems - Antonios Zagaris
Many physical problems are modeled by large systems of nonlinear ordinary differential equations (ODE) with multiple time scales. Typically, the fastest modes are exhausted after a brief transient period and become dependent on the slower ones. In terms of the state space dynamics, the fast transient dynamics bring the solutions close to lower-dimensional invariant manifolds where the long-term dynamics play out. Reduction methods aim to identify such manifolds and to determine the reduced system dynamics on them. This thesis offers analyses for two reduction methods. The first one, the Computational Singular Perturbation (CSP) method, was developed by Lam and Goussis, and it is a method formulated in the tangent bundle of the state space. The second one, the method based on the Zero Derivative Principle (ZDP), was developed by Gear, Kevrekidis, Kaper, and the author, and it is formulated directly in terms of the state space variables.