Nonreciprocity induced spatiotemporal chaos: Reactive vs dissipative routes

TL;DR

This work analyzes a ring of $N$ identical Stuart--Landau oscillators with nonreciprocal left/right coupling to reveal two universal routes to spatiotemporal chaos: reactive coupling with Kerr-type nonlinearity and dissipative coupling with Kerr-type nonlinearity. Through linear stability analysis of the Jacobian and nonlinear simulations, it shows that nonreciprocity broadens growth-rate or eigenfrequency distributions, and that nonlinear saturation converts linear instabilities into chaos, while dissipative nonlinearity suppresses chaotic growth, enforcing bounded periodic states. The results establish a minimal, general framework where nonreciprocity organizes complex dynamics across disciplines and can guide design in photonics, active matter, and ecological systems. Chaos emerges only when the Jacobian eigenvalues form an elliptic distribution, the maximal real part is positive, and Kerr-type nonlinearity is present, highlighting a clear distinction between reactive and dissipative routes.

Abstract

Nonreciprocal interactions fundamentally alter the collective dynamics of nonlinear oscillator networks. Here we investigate Stuart-Landau oscillators on a ring with nonreciprocal reactive or dissipative couplings combined with Kerr-type or dissipative nonlinearities. Through numerical simulations and linear analysis, we uncover two distinct and universal pathways by which enhanced nonreciprocity drives spatiotemporal chaos. Nonreciprocal reactive coupling with Kerr-type nonlinearity amplifies instabilities through growth-rate variations, while nonreciprocal dissipative coupling with Kerr-type nonlinearity broadens eigenfrequency distributions and destroys coherence, which, upon nonlinear saturation, evolve into fully developed chaos. In contrast, dissipative nonlinearities universally suppress chaos, enforcing bounded periodic states. Our findings establish a minimal yet general framework that goes beyond case-specific models and demonstrate that nonreciprocity provides a universal organizing principle for the onset and control of spatiotemporal chaos in oscillator networks and related complex systems.

Figures (7)

• Figure 1: Reactive coupling with Kerr-type nonlinearity. Spatiotemporal patterns of the amplitudes, time series of individual oscillators, and trajectories in the complex plane for a ring of 100 Stuart--Landau oscillators. Parameters are $\mu=-0.5$, $\omega=2.0$, $\xi=0.1i$, and $K=1.0i$. Panels show (a) $J_L=0.75$, (b) $J_L=0.76$, and (c) $J_L=1.0$. In panel (a), the time series and trajectories are plotted from the initial condition, while in (b) and (c) the initial transient has been removed. The diagonal stripes in the amplitude space--time plots indicate that the underlying phase wavefront propagates around the ring at a constant velocity, corresponding to a traveling-wave (rotating-wave) state in (a) and (b).
• Figure 2: (a) Time evolution of the Euclidean norm and (b) complex eigenvalues of Jacobian matrix. The time evolution of the global Euclidean norm, Eq. (\ref{['eq:EucNorm']}), corresponding to the spatiotemporal patterns in Fig. \ref{['fig:fig1']}. The curves illustrate a distinct pathway to spatiotemporal chaos in the reactive--Kerr case. There is transition between decaying behaviors and bounded fluctuation dynamics corresponding to convective decaying states and spatiotemporal chaos, respectively. Parameters are the same as in Fig. \ref{['fig:fig1']}.
• Figure 3: Spatiotemporal patterns of the amplitudes, time series of individual oscillators, and trajectories in the complex plane for a ring of 100 Stuart--Landau oscillators. Parameters are $\mu = -0.5$, $\omega = 2.0$, $\xi = 0.1 i$, and $K = 1.0$. Panels show (a) $J_L = 0.50$, (b) $J_L = 0.515$, and (c) $J_L = 0.55$.
• Figure 4: (a) Time evolution of the Euclidean norm and (b) complex eigenvalues of Jacobian matrix. The time evolution of the global Euclidean norm, Eq. (\ref{['eq:EucNorm']}), corresponding to the spatiotemporal patterns in Fig \ref{['fig:fig3']}. The curves illustrate a distinct pathway to spatiotemporal chaos in the dissipative--Kerr case. There is unbounded dynamics for symmetric case, while bounded fluctuation behaviors corresponding to spatiotemporal chaos increase as asymmetry increases. Parameters are the same as in Fig. \ref{['fig:fig1']}.
• Figure 5: Reactive coupling with dissipative nonlinearity. Spatiotemporal amplitude patterns, oscillator time series, and trajectories in the complex plane for a ring of 100 Stuart--Landau oscillators. Parameters are $\mu=-0.5$, $\omega=2.0$, $\xi=0.1$, and $K=1.0i$. Panels show (a) $J_L=0.75$, (b) $J_L=0.76$, and (c) $J_L=1.0$. The amplitude plots show traveling waves that evolve into stable limit cycles. Here, nonlinear damping immediately balances growth, preventing chaos and enforcing bounded periodic oscillations.