Synchronization facilitated by frequency differences: Dynamics of coupled-oscillator systems with damaged elements

TL;DR

The paper investigates synchronization in globally coupled oscillator populations containing damaged elements modeled as damped oscillators. Using a three-population Stuart-Landau model and a linear-stability analysis of the fixed point, it reveals a reentrant synchronization phenomenon: increasing the frequency difference $\Delta \omega$ first desynchronizes and then re-synchronizes the system due to the damped subpopulation, with boundaries $p_i(\Delta \Omega)$ and $p_{ii}(\Delta \Omega)$ governed by Hopf instabilities. The authors show that symmetry in the frequency distribution yields two Hopf boundaries that merge under certain conditions, and breaking symmetry splits these boundaries, yet a reentrant region persists, indicating the effect is robust and general for near-Hopf systems. The work highlights the damped-element fraction $p$ as a controllable parameter to induce or enhance global synchronization in networks with damaged components, with broad relevance to physics, biology, and engineering.

Abstract

This study investigates the synchronization dynamics of coupled-oscillator systems in which some of the oscillators are damaged and lose their autonomous oscillations. The damaged elements are modeled using damped oscillators; thus, the system is composed of both limit-cycle oscillators and damped oscillators. In this system, as is commonly observed in conventional coupled limit-cycle oscillators, synchronization among oscillators is destroyed when the difference between the natural frequencies of the oscillators increases. However, in the presence of damped oscillators, synchronization can be facilitated by further increasing the frequency difference from the desynchronization state. We conduct numerical simulations on coupled Stuart-Landau oscillators and investigate this reentrance of synchronization systematically. We also propose an approximate theory to predict the stability of the synchronization state based on a linear stability analysis of the fixed point, which reveals the appearance of the Hopf modes. Using this theory, we argue that the reentrance of synchronization driven by increasing frequency differences can be observed in a wide range of coupled-oscillator systems with damaged elements.

Figures (4)

• Figure 1: Phase diagram for synchronization, desynchronization, and amplitude death depending on frequency difference $\Delta \omega = \omega_{A2} - \omega_{A1}$ and the population ratio $p$ of inactive oscillator. Blue circles, red crosses, and green triangles plot respectively the parameter values of synchronization, desynchronization, and amplitude death identified from numerical simulations. Black curves indicate theoretical predictions of transition boundaries (see Section IV). (a) $\omega_{A1} = 5.0, \omega_{A2} = 5.0 + \Delta \omega$ and $\omega_{I} = 5.0$. The time series of each oscillator for $p=0.7$ are shown in the insets to illustrate the typical dynamics. Real part of $A_1, A_2$ and $I$ are plotted respectively by blue, red, and black curves. Frequency differences are respectively set as $\Delta \omega = 0.4, 2.0$ and $3.2$. Parameter changes between them are highlighted in the phase diagram by a gray arrow. (b) $\omega_{A1} = 5.0, \omega_{A2} = 5.0 + \Delta \omega$ and $\omega_{I} = 4.0$. (c) $\omega_{A1} = 5.0, \omega_{A2} = 5.0 + \Delta \omega$ and $\omega_{I} = 6.0$. Other parameters are set as $\alpha = 1.0, \beta = 1.0$ and $\gamma_A = \gamma_I = 0$ for all panels.
• Figure 2: (a) Border curves $p_{\textrm{i}}(\Delta \Omega)$ and $p_{\textrm{ii}}(\Delta \Omega)$. (b) Comparison of analytical and numerical results. Blue circles, red crosses, and green triangles plot respectively the parameter values of synchronization, desynchronization, and amplitude death identified from numerical simulations. Black curves indicate the theoretical prediction of transition boundaries obtained by solving a cubic equation (\ref{['eq00']}). Parameters are set as $\omega_1 = \omega_2 = 5.0, K = 1.2, \alpha = 1.0, \beta = 1$ and $\gamma_A = \gamma_I = 0$.
• Figure 3: Formation of the reentrant synchronization region in the phase diagram by breaking the symmetry in the frequency distribution. Blue circles, red crosses, and green triangles plot respectively the parameter values of synchronization, desynchronization, and amplitude death identified from numerical simulations. Black curves indicate the theoretical prediction of transition boundaries obtained by solving a cubic equation (\ref{['eq00']}). Border curves are obtained by solving the cubic equation numerically. Parameters are set as $\omega_1 = 5.0, K = 1.2, \alpha = 1, \beta=1$ and $\gamma_A = \gamma_I = 0$ for all panels. Results for (left) $\omega_2 = 5.0$, (center) $\omega_2 = 5.25$ and (right) $\omega_2 = 5.5$.
• Figure 4: Phase diagrams for (a)$(\beta, \gamma_A, \gamma_I) = (2.0, 0, 0)$, (b)$(\beta, \gamma_A, \gamma_I) = (1.0, 1.0, 1.0)$ and (c)$(\beta, \gamma_A, \gamma_I) = (2.0, 1.0, 1.0)$. Blue circles, red crosses, and green triangles plot respectively the parameter values of synchronization, desynchronization, and amplitude death identified from numerical simulations. Black curves indicate the theoretical prediction of transition boundaries obtained by solving a cubic equation (\ref{['eq00']}). Other parameters are set as $\omega_{A1} = 5.0, \omega_{A2} = 5.0 + \Delta \omega$, $\omega_{I} = 5.0$, $\alpha = 1.0$ and $K=1.2$.