Synchronization with Annealed Disorder and Higher-Harmonic Interactions in Arbitrary Dimensions: When Two Dimensions Are Special

TL;DR

This work analyzes a D-dimensional Kuramoto model with annealed disorder and both fundamental and higher-harmonic couplings. By developing a nontrivial center-manifold reduction for arbitrary D, it reveals that annealed disorder removes the odd–even dimensional synchronization dichotomy observed with quenched disorder, yielding continuous, mean-field-like transitions across dimensions when only the first harmonic is present, with a tunable continuous-to-discontinuous shift introduced by higher harmonics. A novel correlation-driven transition emerges, captured by a second-order moment M̃, which signals symmetry breaking without global synchronization and exhibits a continuous transition in D=2 but a discontinuous one for D>2. The two macroscopic descriptors, P (synchronization) and M̃ (inter-axis correlations), together describe a richer phase diagram, including a dimensional re-emergence of two-dimensional special behavior under higher-harmonic interactions. The results offer a unified analytical framework for high-dimensional synchronization and point to broad implications for disorder, finite-size effects, and potential control strategies in complex oscillator systems.

Abstract

The impact of disorder on collective phenomena depends crucially on whether it is quenched or annealed. In synchronization problems, quenched disorder in higher dimensional Kuramoto models is known to produce unconventional dimensional effects, including a striking odd even dichotomy: synchronization transitions are continuous in even dimensions and discontinuous in odd dimensions. By contrast, the impact of annealed disorder has received comparatively little attention. Here we study a D dimensional Kuramoto model with both fundamental and higher-harmonic interactions under annealed disorder, and develop an arbitrary dimensional center-manifold framework to analyze the nonlinear dynamics near the onset of collective behavior. We show that annealed disorder fundamentally alters the role of dimensionality. With fundamental coupling alone, it completely removes the odd even dichotomy, yielding continuous synchronization transitions with universal mean-field scaling in all dimensions. Higher-harmonic interactions preserve this universality while rendering the synchronization transition tunable between continuous and discontinuous. At the same time, they give rise to a novel, correlation-driven transition between a symmetry-protected incoherent phase and a symmetry broken state lacking global synchronization, which is therefore invisible to the conventional Kuramoto order parameter. This transition is continuous in two dimensions but discontinuous in higher dimensions, revealing an emergent and previously-unrecognized special role of two dimensions.

Figures (4)

• Figure 1: Phase diagram of the model is presented in (a) for $D=2$ and in (b) for $D>2$ in the $K_1$ (strength of the fundamental interaction) -- $K_2$ (strength of the higher harmonic interaction) plane . The red dashed lines denote continuous transition and the blue solid lines denote discontinuous transition. The enclosed region by the blue solid and the red dashed lines and the $K_1$, $K_2$ axes denote the parameter region in which the uniformly incoherent state (c), denoted by $\mathbb{P}=\mathbb{B}=0$, is linearly stable. The horizontal red dashed and the blue solid lines denote the transition points at which the order parameters $\mathbb{P}$ and $\mathbb{M}$ shows a transition to a synchronized phase (d). The verticle red dashed /blue solid line denotes the transition points of the uniformly incoherent state to a symmetry broken state without global order (e) denoted by $\mathbb{P}=0$ and $\mathbb{B}\neq0$.
• Figure 2: The figure shows numerical-integration results for the stationary state $|\vec{\mathbb{P}}|$, i.e., $\mathbb{P}_\mathrm{st}$ versus $K_1$ for $D = 3$ and $4$, respectively for the case with the noise strength $T = 1$ and $K_2=0$. The black dashed line represents the theoretical prediction of the critical points. All simulation data presented in this paper were generated by integrating the equations of motion using a combination of standard fourth-order Runge-Kutta and Euler algorithms, with integration time step equal to $0.001$.
• Figure 3: Figure (a) shows the $N\to \infty$ phase diagram of the model defined in Sec. \ref{['sec: model 1 formulation']} for $D=4$. The red dashed line marks a continuous transition, while the blue solid line indicates a discontinuous transition; at both transitions, the synchronization order parameter $\mathbb{P}_\mathrm{st}$ become nonzero simultaneously. The thick blue vertical solid line denotes a discontinuous transition in $\mathbb{B}_\mathrm{st}$ alone, across which $\mathbb{P}_\mathrm{st}$ remains zero. Panels (b) and (c) show the behavior of numerically obtained $\mathbb{P}_\mathrm{st}$ as a function of $K_1$ for a continuous case $(K_2 = 2.8)$ and a discontinuous case $(K_2=3.2)$, respectively. The noise strength is $T=1.0$. The square and the circular markers correspond to $N=10^3$ and $10^4$ respectively.
• Figure 4: Stationary value of the order parameters $\mathbb{P}_{\mathrm{st}}$ (triangular symbols) and $\mathbb{B}_{\mathrm{st}}$ (circular symbols) are shown as functions of the second–harmonic coupling $K_{2}$ for fixed $K_{1}=1.0=T$ and system size $N=10^{4}$. Results are displayed for (a) $D=2$, (b) $D=3$, and (c) $D=4$. Panels (d) and (e) show typical late–time configurations of the oscillators for $D=2$ at $K_{2}=6.0$ and for $D=3$ at $K_{2}=11.0$, respectively. For $D=4$, panels (f) and (g) display the time evolution of $\mathbb{B}(t)$ for fifteen independent noise realizations at $K_{2}=11.0~(<12.0=K_2^\mathrm{c})$ and $K_{2}=13.0~(>12.0=K_2^\mathrm{c})$, respectively. Panel (h) shows as a function of the system size $N$ at fixed $K_{2}=11.0$ the fraction of realizations (out of a total of forty–five noise realizations) in which the order parameter $\mathbb{B}$ jumps from zero to a nonzero value within an observation time $t=15$.