Geometric perspective of linear stability in finite networks of nonlinear oscillators

TL;DR

Addresses linear stability of finite Kuramoto networks with delays, where continuum-limit analyses may fail. It introduces a complex-valued operator transformation that reduces the nonlinear Kuramoto dynamics to spectral properties of a composite matrix $\mathbf{K}$ encoding connectivity and phase-lag or delays, enabling analytic stability predictions for all $q$-states in circulant networks. It provides explicit stability criteria, e.g. $\lambda_{m,q} = \frac{\hat{H}(q+m) + \hat{H}(q-m)}{2} - \hat{H}(q)$ (or, with delays, $\lambda_{m,q} = \frac{\gamma_{q+m} + \gamma_{q-m}}{2} - \gamma_q$), and shows how the largest real part of eigenvalues determines the dominant spatiotemporal pattern and the largest basin of attraction; the method extends to heterogeneous phase-lags and distance-dependent delays, enabling predictions of when phase synchronization or twisted waves emerge and the corresponding spatial frequency, in agreement with numerical simulations on finite $k$-ring and weighted networks. The results provide a design-oriented framework for controlling spatiotemporal dynamics in finite oscillator networks, with potential relevance to neuroscience-inspired systems and engineered networks.

Abstract

We use a complex-valued transformation of the Kuramoto model to develop an operator-description of the linear stability in finite networks of nonlinear oscillators. This mathematical approach offers analytical predictions for the linear stability of $q$-states, which include phase synchronization ($q = 0$) and waves with different spatial frequencies ($|q| > 0$). This approach seamlessly incorporates the presence of time delays (represented by phase-lags in the coupling). With this, we are able to analytically determine the specific combination of connectivity and time delays (phase-lags) that leads to any given $q$-state to be linearly stable. This approach offers a geometric perspective of linear stability in finite networks in terms of the connectivity and delays (phase-lag), and it opens a path to designing and controlling the spatiotemporal dynamics of individual oscillator networks.

Figures (5)

• Figure 1: Linear stability of $q$-states in finite networks.(a) We consider a network with $N = 21$ oscillators on a k-ring graph with k = 2. We first use our finite approach given by Eq. (\ref{['eq:eigenvalues_stability']}) and plot the maximum eigenvalue ($\mathrm{max}(\lambda)$) for each $q$-state. We observe that $q=3$-state is the first state to be unstable (blue circles). This is in contrast with the continuum approach, which predicts $q=3$-state to be linearly stable (orange square). (b) We then numerically integrate Eq. (\ref{['eq:main_kuramoto']}) and use, as initial condition, $\bm{\theta}^{(3)}$ in addition to a small perturbation. We observe that, after $50,000$ timesteps, the network transitions to a different state, as the similarity measurement $\mathcal{S}(t)^{(3)}$ that varies from $1$ to $0$.
• Figure 2: Network connectivity and stability.(a) We consider k-ring networks and calculate the eigenvalues of the matrix $\bm{K}$. (b) We then use Eq. (\ref{['eq:eigenvalues_cdt_stability']}) to determine the linear stability of the $q$-states based on the eigenvalues $\gamma$. We can observe how the changes in $\gamma$ due to the connectivity change the stability properties. (c) We consider the full range of $k$ from the first neighbors k = 1 to the complete graph (all-to-all) k = 50, where dark blue indicates the linearly stable states and light blues indicate unstable states. (d) We numerically integrate Eq. (\ref{['eq:main_kuramoto']}) using the $q$-state $\bm{\theta}^{(q)}$ in addition to a small perturbation as initial state and then measure the similarity measurement $\mathcal{S}^{(q)}$ at the end of the simulation at $t = 2 \times 10^{5}$. (e) We can analytically determine the connectivity (represented by $k$) to ensure that phase synchronization is the only linearly stable solution among all the $q$-states (black circles, left). To test our predictions, we integrate Eq. (\ref{['eq:main_kuramoto']}) using $\bm{\theta}^{(1)}$ in addition to a small perturbation as the initial state. We then plot the order parameter $r(t)$ in the end of the simulation (color-code, left), which shows that our predictions are correct. We also show examples of the time evolution of the system $r(t)$ for different conditions (right).
• Figure 3: Linear stability of finite networks on weighted graphs.(a) We consider distance-dependent networks, where the connection weight between two nodes decay with their edge distance. The parameter $\alpha$ controls this decay: for $\alpha = 0$, all nodes are connected with the same weight; as $\alpha$ increases, near nodes are connected with stronger weights. (b) We use Eq. (\ref{['eq:eigenvalues_cdt_stability']}) to analytically predict the linear stability of $q$-states for distance-dependent networks with different $\alpha$ values. (c) We observe that our prediction are correct by using numerical simulations for these networks and evaluating the similarity measurements $\mathcal{S}^{(q)}$ at the end of the simulation ($t = 2.5 \times 10^5$).
• Figure 4: Analytical predictions for the linear stability of $q$-states in networks with phase-lags.(a) Here, we consider k-ring networks, where the pattern of connections is binary (top) and phase-lag is given by $\phi_{jk} = \frac{\pi d_{jk}}{\mathrm{k}}$ (bottom), which varies in $[0,\,\pi]$ (color-code) and no phase-lag is considered between two nodes that are not connected (represented in white). (b) As a specific example, we consider the case with $N = 101$ and k = 10, with and without phase-lag in the coupling, and calculate the eigenvalues $\gamma$ of the matrix $\bm{K}$. (c) We then use Eq. (\ref{['eq:eigenvalues_cdt_stability']}) to determine the linear stability of the $q$-states in the case with phase-lags (blue bars) and without phase-lags (orange bars), where we observe different stable $q$-states due to the presence of phase-lags. We then consider the full range of k $\in [1, 50]$ for these networks with heterogeneous phase-lag, (d) where our analytical predictions given by Eq. (\ref{['eq:eigenvalues_cdt_stability']}) match the (e) numerical simulations of Eq. (\ref{['eq:kuramoto_phase_lag']}) for testing the stability of the $q$-states.
• Figure 5: Analytical predictions for the emergence of waves in delayed networks. We consider a k-ring graph with $N = 101$ and k = 50 (global network), $\epsilon = 1$, and different values of conduction speed $\nu$. We use Eq. (\ref{['eq:eigenvalues_cdt_stability']}) to analytically predict the stability of the $q$-states in the delayed system as the conduction speed is varied. The largest eigenvalue $\gamma$ allows us to estimate the stable $q$-state with largest basin of attraction, which gives us a way to predict the spatial frequency of the wave that emerges for a given $\nu$ (black dots). We then numerically integrate the delayed Kuramoto model Eq. (\ref{['eq:kuramoto_delays']}) with random initial conditions and measure the spatial frequency of the dynamics after $t = 1 \times 10^{5}$ timesteps (gray squares). We repeat this procedure for $1,000$ different initial states, where the gray bars represent the standard deviation.