Extracting Qualitative Dynamics from Numerical Experiments

Abstract

The Phase Space is a powerful tool for representing and reasoning about the qualitative behavior of nonlinear dynamical systems. Significant physical phenomena of the dynamical system---periodicity, recurrence, stability and the like---are reflected by outstanding geometric features of the trajectories in the phase space. This paper presents an approach for the automatic reconstruction of the full dynamical behavior from the numerical results by exploiting knowledge of Dynamical Systems Theory and techniques from computational geometry and computer vision. The approach is applied to an important class of dynamical systems, the area-preserving maps, which often arise from the study of Hamiltonian systems.