Time-Delayed Dynamics in Regular Kuramoto Networks with Inertia: Multistability, Traveling Waves, Chimera States, and Transitions to Seizure-Like Activity

TL;DR

The paper addresses how inertia $m$ and time-delayed coupling $\tau$ affect synchronization in a second-order Kuramoto network with local interactions on a regular ring. It combines analytical stability analysis via a master stability function with extensive time-domain simulations to map phase-locked solutions, bifurcations, and complex patterns. Key findings include delay-induced multistability of fully synchronized states, inertia-driven destabilization and narrowed basins, and the emergence of traveling waves and moving turbulent chimera states that expand with inertia, sometimes yielding seizure-like activity. These results advance understanding of synchronization in neural and rotor networks and have implications for modeling information propagation and stability in delay-coupled systems.

Abstract

This study examines the complex interplay between inertia and time delay in regular rotor networks within the framework of the second-order Kuramoto model. By combining analytical and numerical methods, we demonstrate that intrinsic time delays -- arising from finite information transmission speeds - induce multistability among fully synchronized phase-locked states. Unlike systems without inertia, the presence of inertia destabilizes these phase-locked states, reduces their basin of attraction, and gives rise to nonlinear phase-locked dynamics over specific inertia ranges. In addition, we show that time delays promote the emergence of turbulent chimera states, while inertia enhances their spatial extent. Notably, the combined influence of inertia and time delay produces dynamic patterns reminiscent of partial epileptic seizures. These findings provide new insights into synchronization phenomena by revealing how inertia and time delay fundamentally reshape the stability and dynamics of regular rotor networks, with broader implications for neuronal modeling and other complex systems.

Figures (9)

• Figure 1: (a) Analytically stable, and (b) numerically linear phase-locked frequencies $\Omega_f$ corresponding to complete synchrony versus time delay. The numerical results are presented in the forward direction, using parameter continuation initialized from the previous state. Different panels correspond to inertia values: (i) $m=0$, (ii) $m=0.6$, (iii) $m=1$, (iv) $m=5$, and (v) $m=10$. Other parameters: $N=1000$, $P=5$, i.e., $k_i=\langle k \rangle=2P=10$ for $i=1,\dots,N$, $\omega=1$, and $\alpha=1$.
• Figure 2: (i) Time-averaged order parameter in the steady state, (ii) standard deviation of order parameter fluctuations, (iii) mean rotor angular velocity, and (iv) standard deviation of the angular velocity as functions of time delay. Each row corresponds to a different inertia value: (a) $m = 0$, (b) $m = 0.6$, (c) $m = 1$, and (d) $m = 5$. Red dots represent scatter plots of results from 60 distinct initial conditions, while blue dots indicate ensemble averages across these initial conditions, accompanied by standard error bars. Other parameters as in Fig. \ref{['fig1:w-tau']}.
• Figure 3: The probability density function of angular velocities for an inertia value $m=1$. Each panel corresponds to different time delays: (a) $\tau=0$, (b) $\tau=1.8$, (c) $\tau=2.2$, (d) $\tau=3.1$, (e) $\tau=3.6$, (f) $\tau=4.5$, (g) $\tau=5$, and (h) $\tau=6.2$. Other parameters as in Fig. \ref{['fig1:w-tau']}.
• Figure 4: Frequency evolution of the rotors with inertia value $m=1$. Each panel corresponds to different time delays: (a) $\tau=0$, (b) $\tau=1.8$, (c) $\tau=2.2$, (d) $\tau=3.1$, (e) $\tau=3.6$, (f) $\tau=4.5$, (g) $\tau=5$, and (h) $\tau=6.2$ with $\delta=0.001$ in panels (g) and (h). Other parameters as in Fig. \ref{['fig1:w-tau']}.
• Figure 5: Cosine similarity matrices for an inertia value $m=1$. Each panel corresponds to different time delays: (a) $\tau=0$, (b) $\tau=1.8$, (c) $\tau=2.2$, (d) $\tau=3.1$, (e) $\tau=3.6$, (f) $\tau=4.5$, (g) $\tau=5$, and (h) $\tau=6.2$. Other parameters as in Fig. \ref{['fig1:w-tau']}.