A RANSAC-Based ISOMAP for Filiform Manifolds in Nonlinear Dynamical Systems-An Application to Chaos in a Dripping Faucet

Trajectories generated from a chaotic dynamical system are lying on a nonlinear manifold in the state space. Even if the dimensionality of such a manifold is much lower than that of the full state space, we need many state variables to trace a motion on it as far as we remain to employ the original coordinate, so the resulting expression of the dynamics becomes redundant. In the present study, we employ one of the manifold learning algorithms, ISOMAP, to construct a new nonlinear coordinate that globally covers the manifold, which enables us to describe the dynamics on it as a low-dimensional dynamical system. Here, in order to improve the conventional ISOMAP, we propose an approach based on a combination with RANSAC for pruning the misconnected edges in the neighboring graph. We show that a clear deterministic relationship is extracted from time series of a mass-spring model for the chaotic dripping faucet using the proposed method.