Watanabe-Strogatz Invariants in the Liouvillian Dynamics of Coupled Phase Oscillators via the Koopman Framework

Abstract

In dynamical systems, invariants, i.e., constants of motion conserved along the trajectory, play important roles in characterizing the system's dynamical behavior. Recent applications of the Koopman operator framework to nonlinear dynamical systems have provided new insights into the invariants. For a certain class of globally coupled phase oscillators, which serve as models for various synchronization phenomena, Watanabe and Strogatz proved the existence of N-3 invariants in N oscillator systems. In this study, we derive these invariants from an operator-theoretic perspective by exploiting the relation between Liouvillian (Perron-Frobenius) and Koopman descriptions of the dynamics. Exploiting a simple multiplicative property of functions under the action of the Liouvillian and Koopman operators, we explicitly construct a family of functions whose ratios yield the invariants of the underlying dynamics. Our analysis successfully reproduces the full set of N-3 invariants known in Watanabe-Strogatz theory, and offers an alternative spectral perspective. We demonstrate this approach for a well-studied class of phase models, including the Ermentrout-Kopell, pairwise Kuramoto, and higher-order Kuramoto models.

Figures (3)

• Figure 1: Theta model with $I(t) = \sin t$. (a) Results for $N=3$ oscillators. (a-1) Evolution of the distribution of the phases ${\bm \theta}_3 = (\theta_1, \theta_2, \theta_3)$ on the $(\theta_1-\theta_2) - (\theta_2-\theta_3)$ plane, obtained by simulating the Theta model from $10^5$ initial conditions sampled randomly with the weights proportional to $\left|\psi^3_{\bm{q}}({\bm \theta}_3)\right|$ (clipped at $10^2$). Each black point corresponds to a point ${\bm \theta}_3$ in the state space. (a-2) Density plot of $\left|\psi^3_{\bm{q}}({\bm \theta}_3)\right|$ on the same plane, which is the stationary solution of the Liouville equation, clipped at $10^2$. Vertical, horizontal, and diagonal lines correspond to $\theta_1=\theta_2$, $\theta_2=\theta_3$, and $\theta_3=\theta_1$, which are singular points of $\left|\psi^3_{\bm{q}}({\bm \theta}_3)\right|$ and also plotted in black. (b) Results for $N=10$ oscillators. Typical time evolution of the phases ${\bm \theta}_{10}(t) = (\theta_1(t), \theta_2(t), \ldots, \theta_{10}(t))$ (b-1), $\left|\psi^{10}_{\bm{q}}(\bm{\theta}_{10}(t))\right|$ (b-2), and $\left|\Psi^{10}_{\bm{q}_1,\bm{q}_2}(\bm{\theta}_{10}(t))\right|$ (b-3) with time $t$.
• Figure 2: Kuramoto--Sakaguchi model with $K = 1$ and $\omega = 0$. (a) Results for $N=3$ oscillators with $\delta = \pi/2$. (a-1) Evolution of the distribution of the phases ${\bm \theta}_3 = (\theta_1, \theta_2, \theta_3)$ on the $(\theta_1-\theta_2) - (\theta_2-\theta_3)$ plane, obtained by simulating the Kuramoto--Sakaguchi model from $10^5$ initial conditions randomly sampled as before. (a-2) Stationary state density $\left|\psi^3_{\bm q}({\bm \theta}_3)\right|$ (clipped at $10^2$) of the Liouville equation. (b) Results for $N=10$ oscillators with $\delta = \pi/2$. Evolution of the phases ${\bm \theta}_{10}(t) = (\theta_1(t), \theta_2(t), \ldots, \theta_{10}(t))$ (b-1), $\left|\psi^{10}_{\bm{q}}({\bm \theta}_{10}(t))\right|$ (b-2), and $\left|\Psi^{10}_{\bm{q}_1,\bm{q}_2}({\bm \theta}_{10}(t))\right|$ (b-3) with time $t$. (c) Results for $N=3$ oscillators with $\delta = 0$. (c-1) Evolution of the distribution of the phases ${\bm \theta}_3 = (\theta_1, \theta_2, \theta_3)$ obtained by simulating the Kuramoto--Sakaguchi model from $10^5$ initial conditions randomly sampled with weights proportional to $\left|\psi^3_{\bm{q}}({\bm \theta}_3)\right|$ (clipped at $10^2$). Each black point corresponds to a point ${\bm \theta}_3$ in the state space. (c-2) Evolution of the solution $\left|\psi^3_{\bm{q}}({\bm \theta}_3)\right|e^{-Kt}$ (clipped at $10^2$) of the Liouville equation. On the vertical, horizontal, and diagonal lines satisfying $\theta_1 = \theta_2$, $\theta_2 = \theta_3$, and $\theta_3 = \theta_1$, respectively, the solution diverges but is also plotted in black. (d) Results for $N=10$ oscillators with $\delta = 0$. Evolution of the phases ${\bm \theta}_{10}(t)= (\theta_1(t), \theta_2(t), \ldots, \theta_{10}(t))$ (d-1), $\left|\psi^{10}_{\bm{q}}({\bm \theta}_{10}(t))\right|$ for (d-2), and $\left|\Psi^{10}_{\bm{q}_1,\bm{q}_2}({\bm \theta}_{10}(t))\right|$ (d-3) with time $t$.
• Figure 3: Higher-order Kuramoto model with $K=1$, $\omega = 0$. (a) Results for $N=3$ with $\delta = \pi/2$. (a-1) Evolution of the distribution of the phases ${\bm \theta}_3 = (\theta_1, \theta_2, \theta_3)$ plotted on the $(\theta_1-\theta_2) - (\theta_2-\theta_3)$ plane obtained by simulating the higher-order Kuramoto model from $10^5$ random initial conditions sampled with the weights proportional to $\left|\psi^3_{\bm{q}}({\bm \theta}_3)\right|$ (clipped at $10^2$). Each black point corresponds to a point ${\bm \theta}_3$ in the state space. (a-2) Stationary solution of the Liouville equation, $\left|\psi^3_{\bm{q}}({\bm \theta}_3)\right|$ (clipped at $10^2$). As before, $\left|\psi^3_{\bm{q}}({\bm \theta}_3)\right|$ diverges on the vertical, horizontal, and diagonal lines, but is plotted in black. (b) Results for $N=10$ with $\delta=\pi/2$. Time evolution of the phases ${\bm \theta}_{10}(t) = (\theta_1(t), \theta_2(t), \ldots, \theta_{10}(t))$ (b-1), $\left|\psi^{10}_{\bm{q}}({\bm \theta}_{10}(t))\right|$ (b-2), and $\left|\Psi^{10}_{\bm{q}_1,\bm{q}_2}({\bm \theta}_{10}(t))\right|$ (b-3) with time $t$. (c) Results for $N=10$ with $\delta=0$. Time evolution of the phases ${\bm \theta}_{10}(t) = (\theta_1(t), \theta_2(t), \ldots, \theta_{10}(t))$ (c-1), $\left|\psi^{10}_{\bm{q}}({\bm \theta}_{10}(t))\right|$ (c-2), and $\left|\Psi^{10}_{\bm{q}_1,\bm{q}_2}({\bm \theta}_{10}(t))\right|$ (c-3) with time $t$.

Theorems & Definitions (17)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• proof
• Theorem 3
• proof