Manifold learning approach for chaos in the dripping faucet

Dripping water from a faucet is a typical example exhibiting rich nonlinear phenomena. For such a system, the time stamps at which water drops separate from the faucet can be directly observed in real experiments, and the time series of intervals tau(n) between drop separations becomes a subject of analysis. Even if the mass m(n) of a drop at the onset of the nth separation, which is difficult to observe experimentally, exhibits perfectly deterministic dynamics, it may be difficult to obtain the same information about the underlying dynamics from the time series tau(n). This is because the return plot tau(n-1) vs. tau(n) may become a multivalued relation (i.e., it doesn't represent a function describing deterministic dynamics). In this paper, we propose a method to construct a nonlinear coordinate which provides a "surrogate" of the internal state m(n) from the time series of tau(n). Here, a key of the proposed approach is to use ISOMAP, which is a well-known method of manifold learning. We first apply it to the time series of tau(n) generated from the numerical simulation of a phenomenological mass-spring model for the dripping faucet system. It is shown that a clear one-dimensional map is obtained by the proposed approach, whose characteristic quantities such as the Lyapunov exponent, the topological entropy, and the time correlation function coincide with the original dripping faucet system. Furthermore, we also analyze data obtained from real dripping faucet experiments, which also provide promising results.