Linear pattern dynamics in nonlinear threshold systems

Abstract

Complex nonlinear threshold systems frequently show space-time behavior that is difficult to interpret. We describe a technique based upon a Karhunen-Loeve expansion that allows dynamical patterns to be understood as eigenstates of suitably constructed correlation operators. The evolution of space-time patterns can then be viewed in terms of a ``pattern dynamics'' that can be obtained directly from observable data. As an example, we apply our methods to a particular threshold system to forecast the evolution of patterns of observed activity. Finally, we perform statistical tests to measure the quality of the forecasts.