Effects of Non-reciprocity on Coupled Kuramoto Oscillators

TL;DR

This work demonstrates that non-reciprocal couplings in Kuramoto oscillator networks fundamentally remodel chimera states and generate new time-dependent phases. By analyzing a two-population model and a spatially extended ring, the authors combine numerical simulations with mean-field (Ott–Antonsen) reductions to map phase diagrams in coupling strength and non-reciprocity, revealing run-and-chase, traveling chimera, and chimera II states not present in reciprocal setups. The results show that non-reciprocity shifts transition points, induces reentrant behavior, and yields rich spatiotemporal patterns with potential experimental realizations in artificial oscillator arrays. The study offers a framework for controlling chimera states via directed interactions and motivates future work on higher-dimensional and active-matter oscillator systems.

Abstract

All the fundamental interactions (such as gravity or electromagnetic interactions) are reciprocal in nature. However, in the macroscopic world, in particular outside equilibrium, non-reciprocal or non-mutual interactions are quite ubiquitous. Understanding the impact of such non-reciprocal interactions has drawn a significant amount of interest in physics and other fields of sciences in recent years. We explore a non-reciprocal version of coupled oscillators (known as the Kuramoto model) with the aim of understanding the role of non-reciprocity, particularly in relation to chimera states, where oscillators spontaneously break into mutually synchronous and asynchronous groups. Our findings suggest that non-reciprocity not only alters the state diagram of the chimera state significantly but can also lead to new dynamical states, such as traveling chimera, run-and-chase and coexistence phases.

Figures (11)

• Figure 1: A schematic of our model showing two populations of oscillators with all-to-all coupling. The intra-population coupling has strength $K_0$ whereas non-reciprocal interaction have been implemented in the inter-population coupling as $K^{\rm NR}_{12}=K + \delta$ and $K^{\rm NR}_{21}=K - \delta$.
• Figure 2: (a) Order parameter $\langle m \rangle$ showing a transition from the synchronized state to the stable chimera for $\delta = 0$, at a critical value $A_{c1} = 0.18$. (b) Similar plot for $\delta=0.005$, where the transition occurs at $A_{c1} = 0.118$. The blue circles represent simulation data and the dashed line is a fit to the functional form $\langle m \rangle - m_0 \propto |A-A_{c1}|^{\gamma}$, where $m_{0}$ is the order parameter just above the transition and $\gamma$ is the scaling exponent.
• Figure 3: (a) Order parameter $\langle m \rangle$ showing a transition at $A_{c2}=0.353$ from the breathing chimera to the synchronized state for $\delta = 0$. (b) A similar plot for $\delta=0.005$ shows a transition at $A_{c2} = 0.39$. The red squares are data points obtained from simulations and the dashed line is a fit to the functional form $m - m_0 \propto |A-A_{c2}|^{\gamma}$.
• Figure 4: (a) Mean phase $\theta_\sigma$ ($\sigma \in 1,2$) for the reciprocal case, $\delta = 0$ shows the phase difference ($\Theta$) between the two populations to be zero. (b) For the non-reciprocal case, $\delta=0.6$, though the oscillators in each population have individually synchronized with each other, there exists a finite phase difference $\Theta$, between the oscillators of the two populations. (c,d) show the comparison between the analytical prediction (black dashed line) and the simulation results (symbols) for different $\delta, \alpha$ and $A$.
• Figure 5: Phase diagram of the stability of states in the $(A,\delta)$ plane, constructed by running simulations with $\alpha=1.47$. The value of $\delta$ is represented on the $x$-axis in terms of $\Delta = \mathop{\mathrm{sgn}}\nolimits(\delta) \ln(1 + \frac{|\delta|}{\epsilon})$ with scale parameter $\epsilon= 0.003$. A saddle-node bifurcation occurs at the transition from the run-and-chase phase to the stable chimera phase (red dashed lines) and the transition from stable chimera to breathing chimera is a Hopf bifurcation (black dashed-dot lines). A homoclinic bifurcation occurs when the breathing chimera transitions into the run-and-chase phase (marked by the blue dashed line). All the bifurcation lines are obtained theoretically; see text for details.