Kuramoto meets Koopman: Constants of motion, symmetries, and network motifs

TL;DR

This work reframes Kuramoto oscillators on complex networks through the Koopman operator, yielding rigorous conditions for conserved quantities. It identifies two primary invariant families—monomial eigenfunctions and cross-ratios—and connects their existence to concrete network motifs, including source-driven subgraphs and symmetrizable weight structures. By deriving a Lie-symmetry criterion in the Koopman setting and showing how symmetries generate new constants of motion, the paper provides a path to partial integrability and dimensionality reduction for nonlinear oscillator networks. The approach is validated with analytical motifs, conformist–contrarian dynamics, and empirical-network analyses, revealing a surprisingly prevalent presence of invariants and offering design principles for preserving phase relations in complex systems. The supplementary material extends the theory with detailed proofs, WS reductions, and practical detection methods for motifs in large networks.

Abstract

Partial integrability in phase-oscillator dynamics is typically examined for identically connected oscillators or groups thereof. Yet, the precise connectivity conditions that ensure conserved quantities on general networks remain unclear. Using Koopman theory, we rigorously derive conditions for the existence of distinct conserved quantities in the Kuramoto model with heterogeneous phase lags on any weighted, directed, and signed graph. In the process, we establish a criterion for Lie symmetries in general dynamical systems in terms of the Koopman generator, allowing us to identify symmetries that generate new conserved quantities. These results reveal a broad class of network motifs that support conserved quantities and we detect these motifs in hundreds of complex empirical networks. Our results provide a rigorous theoretical application of Koopman's framework to nonlinear dynamics on complex networks and reveal new possibilities for preserving phase relationships over time in complex systems through a careful design of their structure.

Figures (10)

• Figure 1: The Kuramoto model, its constants of motion and its Lie symmetries under Koopman's perspective. (a) The complex weight matrix $A$ in Eq. \ref{['eq:complex_matrix']} encodes all the parameters of any network of Kuramoto oscillators, whose evolution is described by $\bm{z} = (z_1,...,z_N)$ on the $N$-torus. Observables $f$ for the model are complex-valued functions on the $N$-torus, possibly time-dependent. The observable $(1/N)\sum_{j=1}^Nz_j$ lies in the closed unit disk $\mathbb{D}$ and its modulus, the Kuramoto order parameter Schroder2017, measures synchronization. (b) The observables belong to a function space $\mathcal{O}$. The Koopman generator $\mathcal{K}$ is the total derivative d/d$t$ that generates the time evolution of the observables through the Koopman operator $\exp(t\mathcal{K})$, which composes the observables with the flow $\phi_t$ of the dynamics. An eigenfunction $\psi$ of $\mathcal{K}$ (e.g., monomials in Thm. \ref{['thm:monomials']}) gives key information about the dynamics (e.g., isostables Mauroy2013 as level sets of $|\psi|$). Notably, an eigenfunction with eigenvalue 0 is a constant of motion $C$ (e.g., cross-ratios in Thm. \ref{['thm:crossratios']}). The existence of an eigenfunction $\psi$ with eigenvalue $\lambda$ directly provides a constant of motion $C = \psi e^{-\lambda t}$. (c) A Lie symmetry transforms a solution of the Kuramoto model to a new solution. A transformation is a symmetry provided that an infinitesimal criterion is satisfied: a symmetry generator $\mathcal{S}$ commutes with the Koopman generator, under specific conditions. Indeed, the general form of the criterion is slightly more subtle and is provided in Eq. \ref{['eq:symkoo']}.
• Figure 2: Directed, weighted, and signed network of Kuramoto oscillators with motifs supporting monomial eigenfunctions, conserved cross-ratios, and symmetry-generated constants of motion. (a) Examples for Thm. \ref{['thm:monomials']}. (a1) A single source vertex trivially satisfies all four conditions of Thm. \ref{['thm:monomials']}. (a2) Two oscillators satisfying \ref{['itm:1.1']} and \ref{['itm:1.2']} always satisfy \ref{['itm:1.3']}. (a3) A 5-cycle illustrating \ref{['itm:1.3']}. (a4) A more complex motif yielding a monomial eigenfunction. To construct such a motif, especially to satisfy \ref{['itm:1.3']}, one can define a symmetric matrix and multiply it by a real, nonzero diagonal one to obtain a symmetrizable matrix (\ref{['itm:1.2']}, \ref{['itm:1.3']}). (a5) A source complete graph of oscillators with antisymmetric phase-lag matrix induces a monomial eigenfunction. (b) Examples for Thm. \ref{['thm:crossratios']}. (b1) Eight globally coupled identical oscillators lead to $8 - 3 = 5$ conserved cross-ratios. If these cross-ratios are to coexist with a monomial, the phase-lag matrix is null. If $\alpha_{jk} = \pi/2$, there are $8 - 2 = 6$ WS integrals. (b2) An empty subgraph (\ref{['itm:2.1']}) of oscillators with identical natural frequencies (\ref{['itm:2.3']}) yields a conserved cross-ratio. It can have incoming edges (\ref{['itm:2.2']}) and influence the time evolution of other oscillators (no restriction on the outgoing edges). (b3) The smallest directed star inducing a conserved cross-ratio. (b4) Motif admitting a conserved cross-ratio where two of its vertices admit a monomial eigenfunction. (b5) A non-complete, non-empty and non-star graph yielding a conserved cross-ratio. (b6) Motif of 5 vertices admitting $5-3 = 2$ functionally independent cross-ratios. (c) Examples for Thm. \ref{['thm:thm3']}. (c1) Four (blue and green) vertices with identical natural frequencies (\ref{['itm:3.2A']}) only receiving from a source with identical weights (\ref{['itm:3.1A']}) and distributed into 3 disjoint parts (\ref{['itm:3.3A']}) admit 3 distinct symmetry generators [Lem. \ref{['lem:sym_KL']}]. Thm. \ref{['thm:thm3']}A implies that symmetry generators acting on a conserved cross-ratio yield new constants of motion. The green separation line is meant to include all vertices below it, forming a disjoint part linked to a symmetry generator. (c2) Four (blue and green) vertices with identical natural frequencies $\omega_s - 2\,\mathrm{Im}(\mathcal{A}_s)$ (\ref{['itm:3.2B']}) receiving one edge from source $s$ with weight $\mathcal{A}_s$ (\ref{['itm:3.1B']}) are distributed in 3 disjoint parts (\ref{['itm:3.3B']}). This yields two conserved cross-ratios and 3 symmetries acting on them to form additional constants of motion. (c3) The subgraph (i.e., the rest of the network) admits a symmetry generator but yields no new constant of motion.
• Figure 3: Illustration of different constants of motion, the action of Lie symmetries, and the synchronization of 101 conformist-contrarian oscillators. (a) Complex-weighted graph for the example. Edges between vertices in $\mathcal{W}_6$ exist with Bernoulli probability $p_\ell$ and have random weights $\mathcal{A}_\ell \in \mathbb{C}$ (Gaussian modulus and phase). The operators $\mathcal{S}_2$, ..., $\mathcal{S}_6$ related to $\mathcal{W}_2$, ..., $\mathcal{W}_6$ form an abelian Lie subalgebra of symmetry generators \ref{['eq:liesym']}. (b) The trajectories for $\theta_1, \theta_6, \theta_7$ are constrained by a plane representing the level set of the conserved quantity $\nu_1\theta_1 + \nu_6\theta_6 + \nu_7\theta_7$ (i.e., the argument of the conserved monomial $z_1^{\nu_1}z_6^{\nu_6}z_7^{\nu_7}$). (c, d) The symmetry $\exp(\varepsilon \mathcal{S})$ with $\mathcal{S} = \mathcal{S}_3 + \mathcal{S}_4 + \mathcal{S}_5$ and $\varepsilon = 5$ acts as a Möbius transformation and sends a solution (black curves) constrained by the invariant sets of $c_{1234} \approx 5.0$ and $\mathcal{S}_2[2\ln c_{2345}] \approx 4.2$ to a new solution (blue and green curves) evolving in other invariant sets ($c_{1234} \approx 1.3$, $\mathcal{S}_2[2\ln c_{2345}] \approx -0.6$). The system is in the reference frame of the source ($\omega_1 = 0$), making it possible to illustrate the level sets entirely. (e) Synchronization transitions measured by the time-averaged module $\langle|Z(t)|\rangle_t$ vs. the coupling $\gamma$ when the oscillators from $\mathcal{W}_6$ are isolated and when they are perturbed ($\mathcal{A}_\mathrm{pert} = 20e^{-i\alpha_s}$ with $\alpha_s \in \mathcal{N}(1, 10^{-4})$). The squares and stars indicate corresponding trajectories in (f) and (g); $(Z(t),\phi(t))$ is obtained by integrating the WS equations of $\mathcal{W}_6$ [SI \ref{['sec:param_fig3']}]. (f) Trajectory of $Z(t)e^{-i\phi(t)}$ when $\mathcal{A}_\mathrm{pert} = 0$ reaches an equilibrium point $|Z(t)| < 1$ (blue curve, frequency synchronization). Under the same conditions, except $\mathcal{A}_\mathrm{pert} = 20e^{-0.892i}$, the order parameter rather reaches a limit cycle (orange curve). (g) Trajectory of $Z(t)e^{-i\phi(t)}$ for $\mathcal{A}_\mathrm{pert} = 0$ characterizing frequency synchronization. Under the same conditions, except $\mathcal{A}_\mathrm{pert} = 20e^{-0.823i}$, phase synchronization is feasible (orange curve reaching $|Z(t)| = 1$).
• Figure 4: Presence of network motifs admitting conserved quantities in empirical networks. (a, b) Examples of motifs admitting constants of motion found in the connectome of the annelid P. dumeriliiVeraszt2025 and the web-based network of US government agencies (California) Kosack2018. The blue arrows oriented toward the motifs in (b) mean that each vertex in the motifs has many incoming edges (satisfying \ref{['itm:2.2']}) from vertices outside the motif. The grey arrows with the ellipses indicate that the vertices have many outgoing edges. (c) Composition of the dataset, with networks mainly from https://networks.skewed.de/ and https://github.com/MATPOWER/matpower/tree/master/data, and number of admitted constants of motion. (d) Histogram of the lower bound on the number of (time-dependent) constants of motion in the 498 networks admitting at least one conserved quantity. We count the number of sources, 2-sources (each associated to a conserved quantity $z^{\bm\mu} e^{\lambda t}$), and for each motif of size $m$ admitting conserved cross-ratios, we count the $m - 3$ functionally independent ones. We exclude the conserved cross-ratios associated to the presence of at least four single sources (similarly for (g)). For undirected networks, there are no monomial eigenfunctions by Thm. \ref{['thm:monomials']} (the undirected empirical networks are of course not empty, not complete, considered connected, and they have no sources). Moreover, the only two motifs that admit conserved cross-ratios by Thm. \ref{['thm:crossratios']} are mutually disconnected vertices in a motif of at least four vertices (forming the periphery of star graphs with one or multiple cores in connected networks) or completely connected motifs of at least four vertices (a maximal clique). (e, f, g) Histograms of the number of sources, 2-sources, and motifs admitting cross-ratios.
• Figure S1: Projection of the different points $z_a, z_b, z_c, z_d \in \mathbb{T}^1\setminus\{z_x\}$ to a line in $\mathbb{C}$ from a point $z_x \in \mathbb{T}^1$.

Theorems & Definitions (63)

• Theorem 1: Monomial eigenfunction
• Theorem 2: Cross-ratio conservation
• Lemma : Infinitesimal condition for Lie symmetries
• Theorem 3: Symmetry-generated constants of motion
• Definition S1
• Remark S2
• Lemma S1
• proof
• Lemma S2
• proof