Turbulent Chimera States in Coupled Systems

• Turbulent chimera states are spatiotemporal regimes in coupled systems characterized by coexisting domains of synchrony and chaotic fluctuations.
• Mathematical models—including Stuart–Landau oscillators, nonlinear Schrödinger equations, and the complex Ginzburg–Landau equation—quantitatively capture bifurcation routes and instabilities leading to these states.
• Experimental and numerical studies in laser arrays, oscillator chains, and dissipative lattices validate the existence of turbulent chimeras and offer insights for controlling localized chaos.

Turbulent chimera states are spatiotemporal regimes observed in coupled oscillatory systems—such as networks of phase oscillators, continuous media, and dissipative lattices—in which clusters of synchronized or coherent elements coexist with domains exhibiting high-dimensional chaotic or irregular dynamics. Unlike classical chimeras, which possess stationary or periodically modulated boundaries between order and disorder, turbulent chimeras manifest aperiodic, erratic fluctuations in both the spatial extent and the location of coherent domains. These states arise generically in systems with appropriate bifurcation or instability structures and are characterized quantitatively by irregular variations of coherence measures and positive Lyapunov exponents. Their realization spans a broad spectrum of physical platforms, from globally coupled oscillator arrays and semiconductor laser chains to nonlocal oscillator networks and continuous media governed by partial differential equations.

1. Mathematical Models and Key Formulations

Turbulent chimera states have been analyzed in several mathematical contexts:

• Globally Coupled Oscillators: The Stuart–Landau ensemble described by

$$\frac{dW_k}{dt} = W_k - (1 + i c_2)|W_k|^2 W_k - (1 + i \nu)\langle W\rangle + (1 + i c_2)\langle|W|^2 W\rangle,$$

where $\langle\cdot\rangle$ denotes averaging over the population and $W_k \in \mathbb{C}$ (Haugland et al., 2021). Chimera and turbulent chimera regimes are tracked using order parameters such as the Kuramoto mean field and cross-correlation algorithms.

• Parametrically Driven Dissipative Lattices: A discrete damped nonlinear Schrödinger equation,

$$\frac{dA^j}{dt} = -i \epsilon (A^{j+1} + A^{j-1} - 2A^j) + \gamma (A^j)^* - i \nu A^j - \alpha A^j - i |A^j|^2A^j,$$

supports localized turbulent chimeras characterized by $\lambda_{\max} > 0$ and broadband spectra (Cabanas et al., 2021).

• Nonlocally Coupled Phase Oscillator Chains: Ott–Antonsen and mean-field reductions reveal bifurcation cascades from self-organized quasiperiodic states to spatiotemporal turbulence (Bordyugov et al., 2010).
• Laser Arrays and Continuous Media: Large arrays of quantum-well lasers, modeled via coupled amplitude-phase equations with significant amplitude–phase coupling, display turbulent chimeras in specific parameter bands of coupling strength and detuning (Shena et al., 2016). In continuous media, the complex Ginzburg–Landau equation demonstrates frozen vortex chimeras as a result of purely local diffusive coupling and coupling-field fluctuations (Nicolaou et al., 2017).

These models provide the framework for exploring cascade mechanisms, stability analysis, and the quantitative classification of turbulence in chimera states.

2. Bifurcation and Transition Mechanisms

The transition to turbulence in chimera states typically involves a sequence of bifurcations and attractor interactions:

• Cluster-Splitting Cascade: Globally coupled systems evolve from synchrony through equivariant Hopf and successive period-doubling bifurcations, resulting in a hierarchy of cluster configurations. At each stage, the smallest synchronized cluster splits, eventually leading to configurations with only a single large cluster and many singleton oscillators (Haugland et al., 2021).
• Attractor Collisions and Symmetry-Increasing Bifurcations: Beyond certain torus bifurcations, attractors representing different cluster arrangements collide, increasing symmetry and destroying clusters—this initiates turbulent dynamics. Quantitatively, the transition to chaos aligns with the emergence of positive Lyapunov exponents (e.g., in Stuart–Landau systems, $\lambda_{\max} \sim 10^{-2}-10^{-1}$ at critical parameter boundaries).
• Modulational Instabilities: In dissipative lattices, modulational instability of continuous-wave backgrounds triggers the formation of leaping-wave localized patterns, which can undergo further instability to produce quasi-periodic and then chaotic (turbulent) confined chimeras as control parameters (e.g., drive strength $\gamma$) are increased (Cabanas et al., 2021).
• Instabilities in Coupled Rotators with Inertia: For coupled populations of rotators, an increase in inertia parameter $m$ prompts a bifurcation from nonchaotic breathing chimeras to intermittent chaotic (turbulent) chimeras, where one population remains locked and the other exhibits high-dimensional aperiodic fluctuations. The turbulent regime is marked by a Lyapunov exponent $\Lambda^*$ converging to a positive value in the thermodynamic limit (Olmi, 2015).

These mechanisms underpin the formation of high-dimensional attractors manifest as turbulent chimeras, linking classical bifurcation theory to spatiotemporal complexity.

3. Quantitative Classification and Diagnostic Criteria

The identification of turbulent chimeras relies on rigorous measures of coherence and chaos:

• Local Curvature/Order Parameter ($g_0(t)$): Based on discrete Laplacian or spectral filtering of node amplitudes, $g_0(t)$ quantifies the fraction of spatially coherent elements. The hallmark of turbulence is highly irregular, aperiodic temporal evolution of $g_0(t)$, distinguishing it from stationary (constant) or breathing (periodic) chimeras (Kemeth et al., 2016, Shena et al., 2016). In laser arrays, $g_0(t)$ fluctuates erratically between approximately 0.2 and 0.8.
• Temporal Correlation ($h_0$): Calculated from Pearson-type correlation matrices, $h_0$ distinguishes static from moving turbulent chimeras, i.e., whether coherent domains persistently occupy certain sites or wander through the medium (Kemeth et al., 2016).
• Largest Lyapunov Exponent ($\lambda_{\max}$): $\lambda_{\max} > 0$ serves as a definitive marker of chaos; stationary and quasi-periodic chimeras have $\lambda_{\max} \leq 0$, whereas turbulent chimeras show substantial positivity (e.g., $\lambda_{\max} \sim 0.05-0.1$ for discrete lattices).
• Power Spectra: Turbulent chimeras exhibit broadband, continuous power spectral densities of global norms or local amplitudes; quasi-periodic states produce discrete peaks.
• Statistical Measures: In oscillator chains, the distributions of spatial patch lengths and lifetimes are typically exponential, and autocorrelation functions of local order parameters decay rapidly, indicating spatiotemporal intermittency.

The application of these quantitative diagnostics enables rigorous classification across diverse physical realizations.

4. Physical Realizations and Spatiotemporal Features

Turbulent chimeras have been documented in several experimental and theoretical contexts:

• Semiconductor Laser Arrays: Turbulent chimeras emerge at intermediate values of evanescent coupling strength and linearly graded detuning, notable for irregular dynamics of the coherent domain fraction $g_0(t)$ and rapid migration of spatial coherence patches (Shena et al., 2016). This is attributed to the nonlinear interplay between phase-locking and detuning-induced incoherence, with amplitude–phase coupling amplifying instabilities.
• Oscillator Chains and Networks: Nonlocally and nonlinearly forced oscillator chains show self-emerging chimeras that lose stability to turbulent regimes upon crossing critical system lengths ($L$), with persistent moving and merging patches of local synchrony (Bordyugov et al., 2010).
• Parametrically Driven Lattices: Spatially confined turbulent chimeras in dissipative lattices arise from modulational instability, with chaos contained by steep front solutions. Applications are suggested for nonlinear optics waveguide arrays and cold-atom lattices (Cabanas et al., 2021).
• Continuous Media: The two-dimensional complex Ginzburg–Landau equation, under strictly local coupling, supports “frozen vortex chimeras”—a stationary spiral core surrounded by amplitude turbulence. Fluctuations of the local coupling field, not vanishing even in the continuum limit, restrict the coherent domain size, separating this class from mean-field discretized networks (Nicolaou et al., 2017).

These realizations emphasize the generality of turbulent chimera phenomena, demonstrating their relevance across oscillator systems, pattern-forming media, and quantum devices.

5. Stability Analysis, Transience, and Scaling Properties

Stability properties and the persistence of turbulent chimeras are nontrivial:

• Finite vs Thermodynamic Limit: In coupled rotator populations, intermittent chaotic chimeras are generally transient for finite $N$ but become stationary and persistent as $N \to \infty$, with laminar intervals vanishing and turbulence becoming permanent. The average lifetime $\langle \tau \rangle$ diverges with $N$ and inertia $m$, scaling as $\langle \tau \rangle \propto N^\alpha m^\beta$ ($\alpha \sim 1.6$, $\beta = 2-3$) (Olmi, 2015).
• Chaotic Saddle Mechanism: In networks with multistable elements (e.g., Duffing oscillators and logistic maps), turbulent chimeras can arise from dynamical trapping near high-dimensional chaotic saddles, displaying supertransient lifetimes with exponential scaling in network size. Two subtypes are categorized: S-type (sudden collapse to synchrony) and D-type (gradual invasion of desynchrony); both are underpinned by escape dynamics from the saddle (Medeiros et al., 2023).
• Transversal Stability: Master stability function analysis demonstrates that synchronized states are asymptotically stable for all coupling strengths and ranges, with turbulent chimeras and incoherent states occupying coexisting basins of attraction. Basin stability analyses chart the regions of parameter space where chimera states dominate (Medeiros et al., 2023).
• Patch Statistics: In spatiotemporal turbulence, coherent patch lengths and lifetimes display exponential statistics, with mean values diverging near critical dimensions or bifurcation points (Bordyugov et al., 2010).

These observations clarify the role of phase-space structure, network topology, and coupling schemes in governing the persistence and decay of turbulent chimeras.

6. Distinctive Features and Broader Implications

Turbulent chimeras possess distinctive signatures and motivate broad theoretical and applied inquiry:

• Intermittent Coherence: Aperiodic spatiotemporal modulation of synchrony, with coherent patches dynamically emerging, merging, and disappearing.
• Nonstationary Borders: Absence of fixed spatial boundaries between coherent and incoherent regions; continual wandering and reshaping.
• Amplitude Chaos: Genuine amplitude turbulence (not merely phase desynchronization), especially in dissipative lattices and laser arrays.
• Parameter Sensitivity: Turbulent chimera regimes generically lie in narrow transition bands of coupling and detuning parameters, demarcating boundaries between full synchrony and full incoherence (Shena et al., 2016).
• Relevance Across Domains: Phenomena observed in theoretical models extend to technological platforms (lasers, Josephson arrays), pattern-forming chemical systems, and fluid media.

These properties highlight the universality and robustness of turbulent chimera states, suggesting avenues for controlled generation and manipulation of localized chaos in engineered and natural systems. Their theoretical study elucidates bifurcation structures underpinning high-dimensional dynamical behavior, while experimental realizations offer prospects for applications in photonics, condensed matter, and neuroscience.