  Dynamical Systems (DS) are fundamental to the modeling and understanding time
evolving phenomena, and have application in physics, biology and control. As
determining an analytical description of the dynamics is often difficult,
data-driven approaches are preferred for identifying and controlling nonlinear
DS with multiple equilibrium points. Identification of such DS has been treated
largely as a supervised learning problem. Instead, we focus on an unsupervised
learning scenario where we know neither the number nor the type of dynamics. We
propose a Graph-based spectral clustering method that takes advantage of a
velocity-augmented kernel to connect data points belonging to the same
dynamics, while preserving the natural temporal evolution. We study the
eigenvectors and eigenvalues of the Graph Laplacian and show that they form a
set of orthogonal embedding spaces, one for each sub-dynamics. We prove that
there always exist a set of 2-dimensional embedding spaces in which the
sub-dynamics are linear and n-dimensional embedding spaces where they are
quasi-linear. We compare the clustering performance of our algorithm to Kernel
K-Means, Spectral Clustering and Gaussian Mixtures and show that, even when
these algorithms are provided with the correct number of sub-dynamics, they
fail to cluster them correctly. We learn a diffeomorphism from the Laplacian
embedding space to the original space and show that the Laplacian embedding
leads to good reconstruction accuracy and a faster training time through an
exponential decaying loss compared to the state-of-the-art diffeomorphism-based
approaches.
