Understanding the geometry of dynamics: Invariant manifolds and their interactions
Abstract
Dynamical systems theory studies the behaviour of time-dependent processes. For example, simulation of a weather model, starting from weather conditions known today, gives information about the weather tomorrow or a few days in advance. Dynamical systems theory helps to explain the uncertainties in those predictions, or why it is pretty much impossible to forecast the weather more than a week ahead. In fact, dynamical systems theory is a geometric subject: it seeks to identify critical boundaries and appropriate parameter ranges for which certain behaviour can be observed. The beauty of the theory lies in the fact that one can determine and prove many characteristics of such boundaries called invariant manifolds. These are objects that typically do not have explicit mathematical expressions and must be found with advanced numerical techniques. This paper reviews recent developments and illustrates how numerical methods for dynamical systems can stimulate novel theoretical advances in the field.
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