Extracting Qualitative Dynamics from Numerical Experiments

Kenneth Man-kam Yip

The Phase Space is a powerful tool for representing and reasoning about the qualitative behavior of non-linear 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. Successful use of numerical computations to completely explore the dynamics of the phase space depends on the ability to (1) interpret the numerical results, and (2) control the numerical experiments. This paper presents an approach for the automatic reconstruction of the full dynamical behavior from the numerical results. The approach exploits knowledge of Dynamical Systems Theory which, for certain classes of dynamical systems, gives a complete classification of all the possible types of trajectories, and a list of bifurcation rules which govern the way trajectories can fit together in the phase space. These bifurcation rules are analogous to Waltz’s consistency rules used in labeling of line drawings. The approach is applied to an important class of dynamical system: the area-preserving maps, which often arise from the study of Hamiltonian systems. Finally, the paper describes an implemented program which solves the interpretation problem by using techniques from computational geometry and computer vision.

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