Oscillator Chain Model for Multi-Contour Systems With Priority in Conflict Resolution

TL;DR

The paper develops a continuous oscillator-chain model that generalizes Buslaev nets by representing contour-following vehicle clusters as oscillators $e^{i\theta_i}$ with sectorized interactions defined by $\theta_c$. A Kuramoto-inspired synchronization mechanism is incorporated via $\mathcal{S}_C$ and its coupling $K_C$, while conflict effects are captured by deceleration factors $\mathcal{F}_C$ dependent on a proximity measure $\Lambda_{i,i+1}$ and cluster width $\delta_i$. The main contributions include explicit definitions of the conflict proximity $\Lambda_{i,i+1}$, the deceleration $\mathcal{F}_C$, and the synchronization $\mathcal{S}_C$ factors, plus a governing equation $\frac{d\theta_i}{dt} = \omega_i \mathcal{F}_R \mathcal{S}_R \mathcal{F}_L \mathcal{S}_L$. Numerical results reveal metastable synchronization and phase transitions as the interaction-sector width varies, demonstrating rich dynamical behavior that combines Buslaev-net-like patterns with continuous synchronization dynamics. This framework offers a tractable approach to study network traffic dynamics on complex contour-like topologies and can inform control strategies for multi-contour systems with priority in conflict resolution.

Abstract

We propose a novel model of oscillatory chains that generalizes the contour discrete model of Buslaev nets. The model offers a continuous description of conflicts in system dynamics, interpreted as interactions between neighboring oscillators when their phases lie within defined interaction sectors. The size of the interaction sector can be seen as a measure of vehicle density within clusters moving along contours. The model assumes that oscillators can synchronize their dynamics, using concepts inherited from the Kuramoto model, which effectively accounts for the discrete state effects observed in Buslaev nets. The governing equation for oscillator dynamics incorporates four key factors: deceleration caused by conflicts with neighboring oscillators and the synchronization process, which induces additional acceleration or deceleration. Numerical analysis shows that the system exhibits both familiar properties from classic Buslaev nets, such as metastable synchronization, and novel behaviors, including phase transitions as the interaction sector size changes.

Figures (4)

• Figure S1: The analyzed closed chain of unit-radius contours (I) and the transformation of contours into oscillators (II). The left part of (II) illustrates a contour $i$ with a vehicle cluster moving along it. The cluster's size defines a neighborhood $\mathcal{Q}_{\delta_i}$ around the corresponding node, where the cluster presence may cause a conflict with the motion of a vehicle cluster in the neighboring contour $i-1$. The blurred blue line represents the boundary of the fuzzy neighborhood $\mathcal{Q}_{\delta_i}$, characterized by width $\delta_i$. The right part of (II) shows the equivalent representation of this cluster-contour arrangement in terms of oscillator $i$, depicted as the circle $e^{i\theta_i}$ in the complex plane. The phase $\theta_i$, indicated by the blue arrows, corresponds to the center of the moving cluster. The potential interaction sector with the neighboring oscillator, shown as a darkened region, is determined by the neighborhood $\mathcal{Q}_{\delta_i}$. As observed, the interaction sector size $\theta_{c,i}$ and the width $\delta_i$ of $\mathcal{Q}_{\delta_i}$ are related as $\delta_i = 1 -\cos\theta_{c,i}$, with $\theta_{c,i}$ serving as a measure of the vehicle cluster size.
• Figure S2: The measure $\Lambda_{i,i+1}$ quantifying the proximity of an oscillator pair $\{i,i+1\}$ to the ultimate conflict configuration (I) and the form of the deceleration factor $\mathcal{F}_{L,R}(\theta_i,\theta_{i+1})$ as a function of $\Lambda_{i,i+1}$ for several values of its parameter $p$ (Eq. \ref{['eq:2']}) (II). Blue arrows represent oscillators $i$ and $i+1$ as circles $e^{i\theta_i}$ and $e^{i\theta_{i+1}}$ on the complex plane, where $\theta_i$ and $\theta_{i+1}$ denote their phases. The indices $C=L$ and $C=R$ indicate the oscillator (left or right) whose motion deceleration is quantified by the factor $\mathcal{F}_{C}$, respectively.
• Figure S3: Dynamics of oscillators for different values of the interaction sector width $G/2$ (Eq. \ref{['eq:G']}) in the absence of the synchronization mechanism ($\kappa=0$). The upper row shows the full patterns for 150 selected oscillators, while the lower row presents different fragments of pattern 5, treated either as continuous curves $\{\theta_i(t)\}$ (left plot) or as lines $\{\theta_i(t)\pmod{2\pi}\}$ (with a $-\pi$-shift) confined to the region $[-\pi,\pi]$ (right plot). The blue dotted straight lines (for patterns 2 and 6) indicate the oscillator dynamics assuming their phases could vary synchronously. The other model parameters were set to $\Delta_\theta = 0.2$, $\Delta_\omega = 0$, $\Delta_L = 1.0$, $\Delta_R = 0.5$, and $p=4$.
• Figure S4: Illustration of the effect of synchronization on oscillator dynamics near collapse (upper row) and the dynamics of oscillators with different rotation frequencies (lower row). The plots depict trajectory patterns of 150 selected oscillators. The main parameters are shown in the plots, while the other parameters are set to $G = 2$ and $p = 4$.