Analytical approach to subsystem resetting in generalized Kuramoto models

Abstract

Stochastic resetting has emerged as a powerful mechanism for driving systems into nonequilibrium stationary states with tunable properties. While most existing studies focus on global resetting, where all degrees of freedom are simultaneously reset, recent work has shown that resetting only a subset of degrees of freedom (subsystem resetting) can qualitatively alter collective behavior in interacting many-body systems. In this work, we develop a general theoretical framework for analysing subsystem resetting in Kuramoto-type coupled-oscillator systems. Building on a continued-fraction approach, we derive self-consistent equations for the stationary-state order parameter of the non-reset subsystem, applicable to both noisy and noiseless dynamics and to models with arbitrary interaction harmonics. Using this framework, we systematically investigate how the stationary state and phase transitions depend on the resetting rate, the size of the reset subsystem, and the reset configuration. We show that subsystem resetting can shift or even suppress synchronization transitions, and can give rise to nontrivial features such as re-entrant behavior and restructuring of phase boundaries. In specific cases, including the noiseless Kuramoto model with a Lorentzian frequency distribution, our results recover known analytical predictions and extend them to more general settings. These results establish subsystem resetting as a versatile control protocol for engineering collective dynamics in nonequilibrium interacting systems.

Figures (5)

• Figure 1: Phase diagram of the model \ref{['eq: Kuramoto case IV']}: The horizontal line at $K_1=2D$ shows the transition points where $r_1^\mathrm{st}$ shows a transition upon changing $K_1$ at a fixed $K_2$, with the nature being continuous for $K_2\leq 2D$ (red dashed line) and first-order for $2D<K_2\leq 4D$ (blue solid line). The vertical dotted line at $K_2 = 4D$ shows the transition points where $r_2^\mathrm{st}$ shows a transition upon changing $K_2$ at a fixed $K_1$, with the nature being continuous everywhere.
• Figure 2: Results for the noisy Kuramoto model with first-harmonic interaction and identical frequencies (Sec. \ref{['subsec: Iso-noisy-harmonic Bare']}): Agreement between theory (solid lines) and simulations (unfilled markers) in $r^\mathrm{st}_\mathrm{nr}$ versus $K_1$ for $f=0.2$ is shown for reset configuration $r_0=0.0$ (panel (a)), $r_0=0.4$ (panel (b)), and $r_0=1.0$ (panel (c)). In (a), the filled markers denote the theoretically-obtained transition points $K_1^\mathrm{c}(\lambda)$, Eq. \ref{['eq: critical K model 1 a']}. Agreement between the theoretically-obtained stationary-state distribution (black lines) of the phase-angles of the oscillators from the non-reset subsystem (obtained from Eq. \ref{['eq: eq: p of theta nr4']}) and numerically-obtained histogram is shown in (d) -- (i). In each of these plots, $f$ is chosen to be $0.2$. For $r_0=0$, the plots are shown in (d) for $K_1=2.25$ and in (e) for $K_1=2.7$. For $r_0=0.4$, the plots are shown in (f) for $K_1=1.75$ and in (g) for $K_1=2.5$. For $r_0=1.0$, the plots are shown in (h) for $K_1=1.75$ and in (i) for $K_1=2.2$. In figures (d) -- (i), subplot (1) corresponds to $\lambda = 0.001$, (2) to $\lambda = 0.5$, and (3) to $\lambda = 500$. In all simulations reported in the paper, the dynamics is integrated in time using a combination of the fourth-order Runge-Kutta method and the Euler–Maruyama method Debraj, with integration time step chosen to be 0.0005 for $\lambda=500$ and $0.001$ for rest of the $\lambda$ values. The system size is $N=10^4$.
• Figure 3: Results for the noiseless Kuramoto model with first-harmonic interaction and with uniformly-distributed frequencies (Sec. \ref{['subsec: noiseless-harmonic-uniform Bare']}): Agreement between theory (solid lines) and simulations (unfilled markers) in $r^\mathrm{st}_\mathrm{nr}$ versus $K_1$ for $f=0.2$ is shown for reset configuration $r_0=0.0$ (panel (a)), $r_0=0.4$ (panel (b)), and $r_0=1.0$ (panel (c)). In (a), the filled markers denote the theoretically-obtained transition points $K_1^\mathrm{c}(\lambda)$, Eq. \ref{['eq: critical K model 3']}. The system size is $N=10^4$.
• Figure 4: Results for noisy Kuramoto model with first and second harmonic interaction and identical frequencies (Sec. \ref{['subsec: noisy-harmonic-biharmonic Bare']}): Agreement between theory (solid lines) and simulations (open markers) for the stationary order parameter of the non-reset subsystem, $r_{1,\mathrm{nr}}^{\mathrm{st}}$, as a function of $K_1$ for $f=0.2, D=1.0$. Panels (a)–(c) correspond to $K_2<2D$, while panels (d)–(f) correspond to $K_2>2D$. In each case, the reset configuration is $r_0=0.0$ [(a),(d)], $r_0=0.4$ [(b),(e)], and $r_0=1.0$ [(c),(f)]. Filled markers indicate the theoretical transition points $K_1^{\mathrm c}$ given by Eq. \ref{['eq:K1c_general_model_II_C']}. The system size is $N=5\times 10^3$.
• Figure 5: Noisy Kuramoto model with first and second harmonic interaction and identical frequencies (Sec. \ref{['subsec: noisy-harmonic-biharmonic Bare']}):$K_1^{c}(\lambda)$ as a function of $\lambda$ for fixed values of $K_2$, obtained using Eq. \ref{['eq:K1c_general_model_II_C']}. The dash-dotted line indicates $K_1^c (\lambda=0)=2D$, see Fig. \ref{['fig:bi-phdiag']}. The dotted lines correspond to values of $K_2$ for which the bare model shows a continuous transition in $r_{1,\mathrm{nr}}^{\mathrm{st}}$, while the solid lines correspond to the values of $K_2$ for which the bare model shows a first-order transition in $r_{1,\mathrm{nr}}^{\mathrm{st}}$.