Two-dimensional hydrodynamic simulation for synchronization in coupled density oscillators

A density oscillator is a fluid system in which oscillatory flow occurs between different density fluids through the pore connecting them. We investigate the synchronization in coupled density oscillators using two-dimensional hydrodynamic simulation and analyze the stability of the synchronous state based on the phase reduction theory. Our results show that the antiphase, three-phase, and 2-2 partial-in-phase synchronization modes spontaneously appear as stable states in two, three, and four coupled oscillators, respectively. The phase dynamics of coupled density oscillators is interpreted with their sufficiently large first Fourier components of the phase coupling function.

Figure 1

Two-dimensional model of coupled density oscillators. (a) Schematic drawing of coupled density oscillators consisting of $n$ inner containers. The red broken rectangles represent the calculation areas. (b) Details of the calculation area. The hatched area shows the bottom wall of the inner container.

Figure 2

Synchronization in coupled density oscillators. [(a), (b)] $n=2$, [(c)–(e)] $n=3$, and [(f)–(j)] $n=4$. [(a-1)–(j-1)] Time series of $y_{\mathrm{in}}^{\left(i\right)}-{\overline{y}}_{\mathrm{in}}^{\left(i\right)}$, where ${\overline{y}}_{\mathrm{in}}^{\left(i\right)}$ is the time average of $y_{\mathrm{in}}^{\left(i\right)}$. The grey bold solid, red narrow solid, green bold dashed, and blue narrow dashed lines represent $i=1,2,3,$ and 4, respectively. The symmetry of the initial water level $y_{\mathrm{in}}^{\left(i\right)}\left(0\right)$ is shown above each panel. [(a-2)–(j-2)] Corresponding phase differences $\mathrm{\Delta }{\varphi }_{ij}={\varphi }_j-{\varphi }_i$. The red circles, green crosses, and blue triangles represent $\mathrm{\Delta }{\varphi }_{12}, \mathrm{\Delta }{\varphi }_{13}$, and $\mathrm{\Delta }{\varphi }_{14}$, respectively.

Figure 3

Phase description of a density oscillator. (a) Definition of the phase $\varphi$. The black line shows the time series of $y_{\mathrm{out}}-{\overline{y}}_{\mathrm{out}}$ scaled on the left axis, where ${\overline{y}}_{\mathrm{out}}$ is the time average of $y_{\mathrm{out}}$. The red line shows the corresponding phase $\varphi$ scaled on the right axis. (b) Dependence of the phase shift $\mathrm{\Delta }\varphi$ on the perturbation amplitude $\mathrm{\Delta }{y}_{\mathrm{out}}$. The perturbation was introduced at $\varphi =\pi$ in a cycle, where the phase shift was sufficiently large. The slope of the line was determined by the fitting from five points for $\mathrm{\Delta }{y}_{\mathrm{out}}\le 0.0001$. (c) Phase sensitivity function $Z\left(\varphi \right)$ obtained from the perturbation with $\mathrm{\Delta }{y}_{\mathrm{out}}=0.0001$. (d) Periodic perturbation $f\left(\varphi \right)$ to the water level in the outer container. (e) Phase coupling function $\mathrm{\Gamma }\left(\varphi \right)$. The points represent the discrete data calculated from (c) and (d). The line shows the fitting curve by the Fourier series up to sixth order.

Figure 4

Fourier components of $\mathrm{\Gamma }\left(\varphi \right)$ shown in Fig. 3. (a) Fourier cosine coefficient $a_k$. (b) Fourier sine coefficient $b_k$. The signs of $a_k$ and $b_k$ are represented by solid circles (positive) and open circles (negative).

Figure 5

Phase portrait in the ($\mathrm{\Delta }{\varphi }_{12},d\mathrm{\Delta }{\varphi }_{12}/dt$) plane for two coupled oscillators. It indicates that the antiphase mode for $\mathrm{\Delta }{\varphi }_{12}=\pi$ is stable, whereas the in-phase mode for $\mathrm{\Delta }{\varphi }_{12}=0$ is unstable.

Figure 6

Vector field of ($d\mathrm{\Delta }{\varphi }_{12}/dt,d\mathrm{\Delta }{\varphi }_{13}/dt$) in the ($\mathrm{\Delta }{\varphi }_{12},\mathrm{\Delta }{\varphi }_{13}$) plane for three coupled oscillators. The unstable star nodes (square), the unstable saddle points (diamond), and the stable spirals (circle) represent the all-in-phase, partial-in-phase, and three-phase modes, respectively. The phase velocity $\sqrt{{(d\mathrm{\Delta }{\varphi }_{12}/dt)}^2+{(d\mathrm{\Delta }{\varphi }_{13}/dt)}^2}$ in the ($\mathrm{\Delta }{\varphi }_{12},\mathrm{\Delta }{\varphi }_{13}$) plane is shown as a color gradient.

Figure 7

Behavior of the phase differences $\mathrm{\Delta }{\varphi }_{12}, \mathrm{\Delta }{\varphi }_{13}$, and $\mathrm{\Delta }{\varphi }_{14}$ in four coupled oscillators. (a) Time series of the phase differences from the initial state $(\mathrm{\Delta }{\varphi }_{12},\mathrm{\Delta }{\varphi }_{13},\mathrm{\Delta }{\varphi }_{14})=(\pi /3,7\pi /6,8\pi /5)$. (b) Magnified image for $0\le t\le 50$ in (a). (c) $\mathrm{\Delta }{\varphi }_{13}$ and $\mathrm{\Delta }{\varphi }_{24}$ during the same time range as in (b). The red, green, blue, and magenta lines represent $\mathrm{\Delta }{\varphi }_{12}, \mathrm{\Delta }{\varphi }_{13}, \mathrm{\Delta }{\varphi }_{14}$, and $\mathrm{\Delta }{\varphi }_{24}$, respectively. (d) Phase portrait in the ($\mathrm{\Delta }{\varphi }_{12},d\mathrm{\Delta }{\varphi }_{12}/dt$) plane for the four coupled oscillators with two antiphase pairs $(\mathrm{\Delta }{\varphi }_{12},\mathrm{\Delta }{\varphi }_{13},\mathrm{\Delta }{\varphi }_{14})=(\mathrm{\Delta }{\varphi }_{12},\pi ,\mathrm{\Delta }{\varphi }_{12}+\pi )$. It indicates that the 2-2 partial-in-phase mode for $\mathrm{\Delta }{\varphi }_{12}=0$ and $\pi$ is stable, whereas the four-phase mode for $\mathrm{\Delta }{\varphi }_{12}=\pi /2$ and $3\pi /2$ is unstable.

Figure 8

Comparison of the water level in the outer container $y_{\mathrm{out}}$ among the synchronization modes for (a) $n=2$, (b) $n=3$, and (c) $n=4$. [(a-1)–(c-1)] Time series of $y_{\mathrm{out}}-{\overline{y}}_{\mathrm{out}}$, where ${\overline{y}}_{\mathrm{out}}$ is the time average of $y_{\mathrm{out}}$. [(a-2)–(c-2)] Time series of the total distance of the change in the water level $l$. Among the three simulation results indicating the 2-2 partial-in-phase mode in Figs. 2–2, the simulation result of Fig. 2 is adopted.

Figure 9

Fourier components of $Z\left(\varphi \right)$ and $f\left(\varphi \right)$. (a) Fourier cosine coefficient $p_k$ of $Z\left(\varphi \right)$. (b) Fourier sine coefficient $q_k$ of $Z\left(\varphi \right)$. (c) Fourier cosine coefficient $r_k$ of $f\left(\varphi \right)$. (d) Fourier sine coefficient $s_k$ of $f\left(\varphi \right)$.