Transient rebellions in the Kuramoto oscillator: Morse-Smale structural stability and connection graphs of finite 2-shift type

TL;DR

The paper analyzes finite-N, identical-frequency Kuramoto oscillators to reveal how transients toward synchrony unfold through structured heteroclinic rebellions among cluster equilibria. By introducing partitioned coordinates and leveraging S_N-equivariance, it proves that any heteroclinic edge between 1- and 2-cluster states can be realized by 3-cluster dynamics, and shows transversality that yields a Morse–Smale, structurally stable global dynamic on the attractor. The authors construct a connection graph based on fat/slim cluster partitions, prove concatenation and cascading of rebels, and demonstrate both the local and global robustness of the transient dynamics under small perturbations and in extensions to other network motifs. Numerical simulations corroborate the theoretical framework, including swarming rebellions and left/right transition patterns, while the discussion situates these results within broader oscillator theory and comparisons to Stuart–Landau systems.

Abstract

The celebrated 1975 Kuramoto model of $N$ identical oscillators with phase angle vector $\boldsymbol{\vartheta}=(\vartheta_1,\ldots,\vartheta_N)$ and all-to-all coupling reads \begin{equation} \label{*} \dot\vartheta_j\,=\tfrac{1}{N}\sum_{k=1}^N \sin(\vartheta_k-\vartheta_j). \tag{*} \end{equation} Here we have passed to co-rotating coordinates in normalized time scale. The model is highly accessible to rigorous mathematical analysis, and has been studied as a paradigm for effects like total and partial synchronization. Most initial conditions $\boldsymbol{\vartheta}$ lead to total synchronization. The plethora of $2^N-1$ (circles of) partially synchronized states, however, is unstable. The precise behavior of transitions to synchrony seems to have eluded description. In the present paper, we address this gap. By the gradient structure of (*), the global dynamics decompose into equilibria and heteroclinic orbits between them. Except for the extremes of total instability and total synchrony, all equilibria are 2-cluster solutions: their phase angles $\vartheta_j$ attain only two values, with a phase difference of $π$ between them. Any heteroclinic orbit between 2-cluster equilibria, or towards synchrony, can be realized as a 3-cluster rebellion. Cluster rebellions split the smaller, "slim", minority cluster of the source equilibrium and send the rebellious part to join the bigger, "fat", majority cluster. Heteroclinic transversality identifies the Kuramoto model as a structurally stable Morse-Smale system. In particular, heteroclinic orbits can be concatenated in finite time. The options involved in successive cluster rebellions amount to finite symbol sequences of 2-shift type. The paper is dedicated to Professor Yoshiki Kuramoto, with admiration.

Figures (8)

• Figure 2.1: Heteroclinic left and right 3-cluster rebellions $\boldsymbol{\vartheta}_*(t){:}\; \ \boldsymbol{\Theta}_-\leadsto \boldsymbol{\Theta}_+$ ; see theorem \ref{['2']}. Antipodal angles $x_1=x_-$ (green) and $x_2=x_3=x_-+\pi$ (red) of the 2-cluster source equilibrium $\boldsymbol{\Theta}_-$ and the target equilibrium $\boldsymbol{\Theta}_+$ are marked by solid dots on the phase circle $\mathbb{S}^1$, in (a) and (d). Areas of dots refer to cluster size. In (b) and (c): heteroclinic rebellions of cluster $x_2$ (orange), which splits off from the slim cluster at $x_2=x_3$ to join the fat cluster at $x_2=x_1$. See \ref{['J123']} and \ref{['x2pm']}, respectively. Left rebellion of cluster $x_2$ is illustrated in (b), and right rebellion in (c); see \ref{['left']}--\ref{['right']}. Cases (b) and (c) are interchanged under reflections \ref{['O2']}.
• Figure 2.2: Time plot of $n=10$ successive one-man rebellions towards synchrony, concatenated to a heteroclinic orbit $\boldsymbol{\Theta}_*(t){:}\ \boldsymbol{\Theta}_0\leadsto\boldsymbol{\Theta}_n$ . The three cluster phases (vertical) are $x_1$ (fat, green), $x_2$ (one-man rebellions $N_2=1$, orange) and $x_3$ (slim, red) 3-cluster rebellions. Compare figure \ref{['fig21']} for side-switching of the rebellions. Initial cluster sizes are $N_1=11,\ N_2=1,\ N_3=9$ for $N=21$ oscillators, with phases near equilibrium $x_1/\pi=\alpha_1-1= -10/21$ and $x_2/\pi=x_3/\pi=\alpha_1=11/21$, for $t\rightarrow -\infty$. Thickness of trajectories refers to cluster size. Note the nine successive one-man transients (orange) of faster and faster ejection from the slimming red cluster, and convergence to the fattening green cluster. Left transients, as in figure \ref{['fig21']}(b) appear above the red graph of slim $x_2(t)$, and right transients \ref{['fig21']}(c) appear below. In symbols \ref{['sequence']}, $s= +-+--+-+++$. The final tenth "transient" (red, down) vacates the final near-antipodal red slim "cluster" of "last man standing" opposition, entirely, and achieves synchrony, at last. See FieTatraAsh07KuGiOtt2015 and figure \ref{['fig8']} below for related ideas. Note how each one-man rebellion $s_\ell=\pm$ induces parallel drifts by $s_\ell\,\alpha_2\pi=\pm\pi/N$ in the position $x_1$ of the fat majority cluster. See corollary \ref{['corxshift']}.
• Figure 2.3: Phase plots of $n=10$ successive side-switching one-man rebellions towards synchrony, concatenated to a heteroclinic orbit $\boldsymbol{\Theta}_*(t){:}\ \boldsymbol{\Theta}_0\leadsto\boldsymbol{\Theta}_n$ . The plot is based on the same data as figure \ref{['fig22']}, with the same color coding. Phases $x_m\in\mathbb{S}^1$ are drawn on a cylinder $(x_m,t)\in\mathbb{S}^1\times\mathbb{R}$ with vanishing point of perspective at $t = -\infty$. Thin gray lines mark a cylindrically Cartesian grid with angle sectors of width $\pi/4$. Clockwise orange transients, right, follow figure \ref{['fig21']}(c). Counter-clockwise orange transients, left, follow figure \ref{['fig21']}(b). Their tangencies point to the front, in the direction of time. The final tenth "transient" (red, right) vacates the final near-antipodal red slim "cluster" of "last man standing" opposition, entirely, and achieves synchrony, at last.
• Figure 2.4: Connection graph $\mathcal{C}$ for the Kuramoto model \ref{['kurODE']} with $N = 5$ oscillators. Equilibrium vertices $\boldsymbol{\Theta}_J$ are represented by their complementary slim cluster sets $J^c$ of size $|J^c|=N_2=5-|J|$. Note order parameters $0<R=2\alpha-1=1-2/5|J^c|\leq1$, and Morse indices $i=|J^c|$. The 5-bar linkage equilibrium set $\{\boldsymbol{\Theta} \in \boldsymbol{\Xi}\,|\, R=0\}$ is represented by the artificial top vertex $0$. The lowest level $\emptyset$ signifies an empty slim cluster, that is, total synchrony. Directed edges represent heteroclinicity between adjacent Morse levels. In view of cascading and transverse transitivity, directed paths in $\mathcal{C}$ capture all heteroclinicity. In particular, edges are directed downward, towards lower Morse index $i$ and larger order parameter $0 < R \le 1$.
• Figure 2.5: Time plot of a swarming rebellion from a 2-cluster equilibrium $\boldsymbol{\Theta}_-$ at $t \to -\infty$ towards a 2-cluster equilibrium $\boldsymbol{\Theta}_+$ with larger fat cluster size. See theorem \ref{['4']}. The color coding is the same as in figures \ref{['fig22']}--\ref{['fig23']}. Cluster sizes are $N_-=11,\ N_+=20$ for $N=21$ oscillators $\boldsymbol{\vartheta}_*(t)$, moving in $M=11$ clusters. Limiting cluster phases $x_1/\pi=-10/21$ and $x_2/\pi=\ldots=x_{11}/\pi=11/21$ for $t\rightarrow -\infty$ indicate the same source equilibrium $\boldsymbol{\Theta}_-$ as in figures \ref{['fig22']}--\ref{['fig23']}. Of the $M=11$ clusters, $M-2=9$ are a rebellious swarm of one-man "clusters", preserving the ordering by phases. The swarm of nine is parted into $m_*=5$ right and $9-m_*=4$ left rebels. The final phases at target $\boldsymbol{\Theta}_+$ satisfy $x_1/\pi=-11/21$ and $x_{11}/\pi=10/21$.

Theorems & Definitions (14)

• Proposition 1.1: Classification of equilibria
• Theorem 2.1: Angles and monotonicity of equilibrium $2$-clusters
• Theorem 2.2: Realization of $3$-cluster heteroclinic orbits
• Corollary 2.3: Phase shifts
• Theorem 2.4: Transverse heteroclinics
• Corollary 2.5: Morse--Smale structural stability
• Corollary 2.6: Concatenation of heteroclinic orbits
• Corollary 2.7: Connection graph
• Corollary 2.8: Finite $2$-shift of heteroclinic concatenations
• Theorem 2.9: Realization of arbitrary heteroclinic partitions