Decomposition and Clustering for the Visualization of Dynamical Systems

An effective visualization of the global behavior of a dynamical system or a fluid simulation inevitably involves a sort of the partition, or the decomposition of the phase space of the system. This is because, in a generic system there exist uncountably many points having \u22similar\u22 dynamical behavior and if we plot too many of them then typically we end up with a picture carrying no information. See Figure 1, in which 300 (left) and 3,000 (right) different trajectories of the standard map, the most important example of Hamiltonian dynamics, of length 100 are plotted. It easy to imagine that this problem will be more serious in higher dimensions. Thus, a natural way to visualize the global behavior of the systems is to classify the points in the phase space into a relatively small number of clusters, so that each cluster corresponds to a particular dynamical behavior. In following sections, we will explain two ideas for such a clustering. In the next section, we will briefly review the Conley-Morse decomposition, a decomposition of the phase space according to the gradient-like structure of the system. This method does not work fine for conservative dynamics and hence we will discuss another algorithm based on a graph clustering algorithm in the last section. This is a work in progress in the JST CREST project [6]