Coherence enhanced by detrained oscillators: Breaking $π$-reflection symmetry

Abstract

We study a generalized Kuramoto model in which each oscillator carries two coupled phase variables, representing a minimal swarmalator system. Assuming perfect correlation between the intrinsic frequencies associated with each phase variable, we identify a novel dynamic mode characterized by bounded oscillatory motion that breaks the $π$-reflection symmetry. This symmetry breaking enhances global coherence and gives rise to a non-trivial mixed state, marked by distinct degrees of ordering in each variable. Numerical simulations confirm our analytic predictions for the full phase diagram, including the nature of transition. Our results reveal a fundamental mechanism through which detrained (dynamic) oscillators can promote global synchronization, offering broad insights into coupled dynamical systems beyond the classical Kuramoto paradigm.

Figures (5)

• Figure 1: (Color online) Phase diagram in the $(J_-,J_+)$ plane for $\gamma=1$. Purple and red circles represent numerical results from simulations, while solid lines indicate theoretical predictions. The red curve corresponds to the hyperbolic transition line given by Eq. (13).
• Figure 2: (Color online) The $\pi$-reflection symmetry breaking of the average probability distribution function $P_{\omega}^d(X)$ for a detrained (dynamic) oscillator. The data (red open circles) are obtained from numerical simulations with $N=819,200$, using the parameters $J=9$, $K=3$, and $\omega=4.37$$(> aS_{+})$, showing excellent agreement with the theoretical prediction (black solid line) given by Eq. (S59) in SM ref:SM. For comparison, the same data are replotted with respect to $\pi-X$ (green open circles and orange line), clearly demonstrating the breaking of the $\pi$-reflection symmetry: $P_{\omega}^d(X) \neq P_{\omega}^d(\pi-X)$. The inset shows the corresponding results for $J=K=5$ ($J_-=0$), where the $\pi$-reflection symmetry is preserved.
• Figure 3: (Color online) The order parameters $S_{\pm}$ are plotted as functions of $J_+$ for $J_-=4$ and $\gamma=1$. Open red circles and green squares represent numerical data for $S_{+}$ and $S_{-}$, respectively, in excellent agreement with analytical predictions, including transition points and their nature. The order parameter $S_-$ exhibits a discontinuous jump from 0 to 1 at $J_+=0$, and remains nearly constant at 1 thereafter, except for a slight dip beginning around $J_+ = 6$ (see upper inset). At this point, $S_+$ begins to grow continuously. Both $S_{+}$ and $S_{-}$ approach $1$ asymptotically as $J_+\rightarrow\infty$, without crossing. The solid purple line corresponds to the prediction obtained by neglecting dynamic contributions, which clearly deviates from the numerical data. For comparison, the lower inset shows results for the case with $J_-=0$. Vertical lines indicate the boundaries between different regimes: $(0,0)$, $(0,1)$, and $(S_1,S_2)$.
• Figure S1: (Color online) The order parameters $S_{\pm}$ are plotted as functions of $J_{+}$ for fixed $J_{-}=0$ with $\gamma=1$. Open red circles and green squares represent numerical data for $S_{+}$ and $S_{-}$, respectively. Vertical dashed lines at $J_{+}=0$ and $J_{+}=4$ indicate the boundaries between three distinct states: $(S_{+}, S_{-}) = (0,0)$, $(0,1)$, and $(S,1)$. The solid purple line denotes the analytic solution for $S_+$ given by Eq. (\ref{['Seq:Sps_sameJK']}).
• Figure S2: (Color online) The amplitudes $\alpha_{\pm}$ and critical exponents $\beta_{\pm}$ are plotted as functions of $J_{-}$. Open red circles and green squares represent numerical data obtained from a system of size $N=10^5$, using a Lorentzian distribution with width $\gamma=1$. Statistical errors are also indicated. The inset shows $\beta_{\pm}$ as a function of $J_{-}$. The solid red and green lines in the main panel correspond to Eqs. (\ref{['Seq:alphaplus']}) and (\ref{['Seq:alphaminus']}), respectively, evaluated at $\gamma=1$.