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 $n$th 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.

Figure 1

(a) Schematic illustration of the apparatus for the dripping faucet experiment. (b) Snapshots of a high-speed movie showing the separation of a drop. (c) Illustration of the mass-spring model.

Figure 2

(a) Chaotic trajectory of the Kiyono-Fuchikami model. (b) Time series of drop position $x$ and the time interval $\tau$ between drop separations.

Figure 3

(a) Return plot of $m_{n-1}$ vs $m_n$. (b) Return plot of ${\tau }_{n-1}$ vs ${\tau }_n$. Movements of points as the time delay embedding vector ${\mathit{s}}_n=({\tau }_{n-1},{\tau }_n)$ are also depicted.

Figure 4

(a) Bifurcation diagram of the Kiyono-Fuchikami model. Note that the unit of time is rescaled (e.g., $\tau =14$ corresponds to 0.238 sec). (b) Time series of a real dripping faucet experiment.

Figure 5

Neighboring graph $\mathcal{G}$. (a) Schematic illustration of the sample points ($\circ$) and the three-neighboring graph $\mathcal{G}$ (solid line). The heavy solid line is the shortest path between points $i$ and $j$ and the dashed line is the Euclidean distance. (b) 500 sample points ($+$) and four-neighboring graph $\mathcal{G}$ (line) for $v_0=0.1130$. (c) Graph $\mathcal{G}$ for two-band chaos ($v_0=0.1128$). The two clusters B${}_1$, B${}_2$ caused by two-band chaos are connected by the shortest path C.

Figure 6

Results of the application of isomap for Fig. 5b. (a) Configurations of sample points onto the $(u^{\left(1\right)},u^{\left(2\right)})$ plane. (b) Plot of sample points in the $({\tau }_{n-1},{\tau }_n,u_n^{\left(1\right)})$ space. (c) Relation between the new coordinate $u$ and the droplet mass $m$.

Figure 7

(a), (b) Return plots of the new variable $u^{\left(1\right)}$ generated by isomap for Figs. 5b and 5c, respectively.

Figure 8

(a) Relation between the new variable $u$ and the dripping interval $\tau$. (b) Distribution function of dripping intervals $P\left(\tau \right)$ which is derived from Eq. (12). The result from the direct simulation (${10}^6$ droplets) are also plotted by the symbol $+$.

Figure 9

Time correlation function of ${\tau }_n$. The results from $g\left(u\right)$ ($\circ$) for a return plot (500 droplets) and the direct simulation (${10}^6$ droplets) from Eqs. (3) and (4) ($\times$) are plotted. The size of fluctuation for the different return maps is about 30$\%$ around $n=25$. The two curves show the decay for a period-two and a long-period (about 23) oscillation.

Figure 10

Results for our experimental data. (a) Dripping time series ${\tau }_n$, which contains one or two periodic motion around $n=50,170$. (b) First-return plot of $\tau$. It shows a multivalued function in the right region. (c) First-return plot of inner state variable $u$, which suggests that there is a one-dimensional map $g\left(u\right)$. It may have a wavelike pattern in the right region.