Kuramoto Mean Field Game with Intrinsic Frequencies (2509.18000v1)

Abstract: This paper studies a mean field game formulation of the classical Kuramoto model for synchronization. Our model captures the diversity within the population by considering random intrinsic frequencies, which allows us to study the impact of this heterogeneity on synchronization patterns and stability. Our findings contribute insights into the interplay between intrinsic frequency diversity and synchronization dynamics, offering a more realistic understanding of complex systems. The proposed framework has broad applications ranging from coupled oscillators in physics to social dynamics, and serves as a valuable tool for studying networks with distributed intrinsic frequencies.

• The paper introduces a mean field game framework for the Kuramoto model, emphasizing the influence of heterogeneous frequencies on synchronization.
• It derives explicit critical thresholds for phase transitions using fixed point analysis and the Penrose criterion.
• Numerical examples reveal complex bifurcation structures that offer practical insights for applications in physics, biology, and social systems.

Kuramoto Mean Field Game with Intrinsic Frequencies: A Technical Analysis

The paper develops a mean field game (MFG) framework for the Kuramoto synchronization model, incorporating heterogeneous intrinsic frequencies. This extension enables a rigorous paper of the interplay between frequency diversity and synchronization, with implications for both the theory of MFGs and the analysis of large-scale coupled oscillator systems.

Model Formulation

The system consists of a continuum of agents (oscillators) on the torus $T = \mathbb{R}/2\pi\mathbb{Z}$, each with an intrinsic frequency %%%%1%%%% drawn from a probability distribution $g$ on $\mathbb{R}$. The state evolution for an agent with frequency $\omega$ under control $\alpha$ is given by

$dX_t = \alpha_t dt + \omega dt + \sigma dB_t,$

where $\sigma > 0$ is the noise strength. The agent's objective is to minimize the infinite-horizon discounted cost

$$J^{\omega, \mu}(\alpha) = \mathbb{E} \int_0^\infty e^{-\beta t} \left[ \frac{1}{2} \alpha_t^2 + \kappa c(X_t, \mu_t) \right] dt,$$

where $c(x, \mu)$ encodes the mean-field interaction, and $\kappa$ is the coupling strength. The mean-field term is

$$c(x, \mu) = 1 - \cos(x) \int_T \cos(y) \mu(dy, \mathbb{R}) - \sin(x) \int_T \sin(y) \mu(dy, \mathbb{R}),$$

which depends only on the first marginal of $\mu$.

A mean field Nash equilibrium is a flow of measures $(\mu_t)_{t \geq 0}$ on $T \times \mathbb{R}$ such that, for $g$-almost every $\omega$, the conditional law $\mu_t^\omega$ on $T$ is the law of the optimally controlled process, and the second marginal of $\mu_t$ is always $g$.

Stationary Solutions and Phase Transitions

The uniform measure $m \otimes g$ is always a stationary solution, corresponding to the incoherent state. The main technical focus is on the existence and stability of non-uniform (synchronized) stationary solutions, and the identification of critical coupling thresholds for phase transitions.

The analysis reduces the stationary MFG problem to a two-dimensional fixed point problem for the order parameters $(\alpha_1, \alpha_2)$: $$\alpha_1 = \kappa \int_{T \times \mathbb{R}} \cos(y) \nu^\omega_\infty(y) g(d\omega) dy, \quad \alpha_2 = \kappa \int_{T \times \mathbb{R}} \sin(y) \nu^\omega_\infty(y) g(d\omega) dy,$$ where $\nu^\omega_\infty$ is the invariant measure for the optimally controlled process with frequency $\omega$.

For symmetric $g$, the problem further reduces to a scalar fixed point for $\alpha_1$ (with $\alpha_2 = 0$). The critical threshold for the emergence of non-uniform equilibria is given by

$$\kappa_c(g) = \left( \int_{\mathbb{R}} \frac{\gamma \sigma^2 + 2\omega^2}{(\gamma^2 + \omega^2)(\sigma^4 + 4\omega^2)} g(d\omega) \right)^{-1},$$

where $\gamma = \beta + \sigma^2/2$.

Key results:

• For symmetric $g$, non-uniform stationary solutions exist for $\kappa > \kappa_c(g)$.
• The uniform state is locally stable for $\kappa < \kappa_c(g)$ if $g$ has non-negative Fourier transform.
• For certain $g$ (e.g., sum of Diracs), the actual stability threshold can be strictly less than $\kappa_c(g)$, indicating richer bifurcation structure than in the homogeneous case.

Linear Stability and Penrose Criterion

The stability of the incoherent state is analyzed via linearization and Laplace/Fourier analysis. The linearized operator $L$ acting on perturbations of the order parameters is explicitly computed, and its norm in exponentially weighted spaces provides a sufficient condition for stability.

A sharper criterion is obtained via the Penrose condition, involving the holomorphic function

$$P(z) = \int_{\mathbb{R}} \frac{1}{(\gamma + i\omega - z)(\sigma^2/2 + z - i\omega)} g(d\omega),$$

with the Penrose threshold

$$\kappa_P(g) = \inf \left\{ \kappa > 0 : \exists \theta \in \mathbb{R},\ P(i\theta) = 2/\kappa \right\}.$$

For $g$ with positive Fourier transform, $\kappa_P(g) = \kappa_c(g)$, but for multi-Dirac $g$, strict inequality can occur, as demonstrated numerically and analytically.

Technical Implications and Numerical Results

The analysis rigorously establishes the existence of a phase transition in the MFG Kuramoto model with heterogeneous frequencies, generalizing classical results for the homogeneous case. The explicit formulas for critical thresholds enable precise predictions for the onset of synchronization as a function of the frequency distribution, noise, and discounting.

Numerical examples show that for $g$ a symmetric sum of Diracs, the bifurcation diagram can exhibit multiple nontrivial fixed points even below $\kappa_c(g)$, and the Penrose threshold provides a more accurate stability boundary.

Theoretical and Practical Implications

The results provide a comprehensive framework for analyzing synchronization in large populations with heterogeneity, relevant for applications in physics (e.g., power grids), biology (e.g., circadian rhythms), and social systems. The MFG approach allows for the incorporation of control and optimization, extending beyond the classical Kuramoto model.

Theoretically, the work demonstrates that MFGs with non-monotone, non-potential structure can exhibit complex invariant sets, including multiple stationary and possibly periodic orbits, depending on the frequency distribution. The explicit connection to the Penrose criterion links the stability analysis to classical plasma physics and dynamical systems.

Future Directions

Potential extensions include:

• Analysis of time-dependent (non-stationary) solutions and their bifurcations.
• Study of the effect of non-symmetric or heavy-tailed $g$.
• Numerical schemes for high-dimensional fixed point problems in the presence of heterogeneity.
• Applications to networked systems with more general coupling topologies.

Conclusion

The paper provides a mathematically rigorous and technically detailed analysis of the Kuramoto mean field game with intrinsic frequencies, identifying critical thresholds for synchronization, characterizing the stability of incoherent and synchronized states, and revealing the rich structure induced by heterogeneity. The results bridge mean field game theory, synchronization phenomena, and spectral stability analysis, and open avenues for further research in both theory and applications.