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A classification scheme for chimera states
Felix P. Kemeth, Sindre W. Haugland, Lennart Schmidt, Ioannis G. Kevrekidis, Katharina Krischer
TL;DR
Chimera states exhibit diverse forms of coexisting coherence and incoherence that are not fully captured by existing definitions or system-specific schemes. The paper introduces two correlation measures for spatial and temporal coherence, then uses their behavior to define and classify chimeras across numerical and experimental data. The scheme recovers recognized examples, distinguishes stationary and breathing dynamics, and supports classification across general dynamical systems.
Problem
Existing chimera definitions and characterization schemes do not accommodate newer coexistence patterns and are restricted to small classes of systems.
Method
The paper introduces g0(t) and h0 as measures of spatial and temporal coherence for classifying chimera states from simulated or experimental spatio-temporal data.
Results
The measures assign literature examples to chimera classes and provide a general classification applicable beyond phase-oscillator ensembles.
Takeaways & Limitations
The scheme enables simple classification of general chimera states and qualitative and quantitative comparison across experimental and numerical data.
Abstract
from arXiv · showhide
We present a universal characterization scheme for chimera states applicable to both numerical and experimental data sets. The scheme is based on two correlation measures that enable a meaningful definition of chimera states as well as their classification into three categories: stationary, turbulent and breathing. In addition, these categories can be further subdivided according to the time-stationarity of these two measures. We demonstrate that this approach both is consistent with previously recognized chimera states and enables us to classify states as chimeras which have not been categorized as such before. Furthermore, the scheme allows for a qualitative and quantitative comparison of experimental chimeras with chimeras obtained through numerical simulations.
I. INTRODUCTION
Recent chimera patterns differ in how coherence and incoherence appear, exposing limits in the original definition and existing system-specific characterization schemes. The paper therefore proposes two simple, broadly applicable measures and applies them to numerical and experimental data.
- New chimera patterns include spatial disorder with temporal order, amplitude disorder, and variability in coherent-region behavior.
- The original definition of a chimera as coexisting coherent and incoherent oscillator regions does not accommodate these developments.
- Existing characterization approaches are restricted to minimal networks or systems with non-local coupling.
- The proposed two measures provide a simple definition and distinguish chimeras with different coherence properties.
- The approach is independent of coupling scheme and spatial dimension, and is not limited to phase oscillators.
- The paper introduces spatial and temporal correlation measures, applies them to experimental and simulated data, and develops a classification scheme.
A. A measure for correlation in space
The spatial correlation measure uses local curvature or pairwise distances to estimate the relative amount of coherent structure. Its time evolution distinguishes stationary from breathing behavior, while the same framework applies across spatial and non-spatial systems.
- Spatial correlation measure: Local curvature is computed by rescaling and applying a discrete Laplacian to each spatial snapshot.
- Spatial correlation measure: For phase oscillators, the Laplacian maps synchronized regions toward zero curvature and incoherent regions toward finite, fluctuating curvature.
- Spatial correlation measure: A normalized curvature distribution g yields g(|D̂| = 0), measuring the relative size of spatially coherent regions.
- Spatial correlation measure: Coherence is defined relative to each system’s maximum curvature, with points below 1% of that maximum characterized as coherent.
- Classification by temporal evolution: For the Kuramoto example, g0(t) is about 0.3 and time-independent, identifying a chimera with stationary coherence.
- Spatial correlation measure: For globally coupled systems without spatial extent, pairwise Euclidean distances provide a synchrony measure and yield the relative amount of correlated oscillators.
- Classification by temporal evolution: In the two-group model, g0(t) oscillates periodically, identifying breathing behavior and distinguishing it from constant partial synchronization.
B. A measure for correlation in time
Temporal correlations between oscillator time series provide a measure for distinguishing static chimeras and quantifying their time-correlated component. For the Kuramoto example, a peak at |ρ| = 1 identifies a static state, while h0 is compared with the spatial measure g0(t) but does not always represent synchronized-cluster size.
- Pairwise correlation coefficients between oscillator time series define a measure for correlation in time.The coefficients use the oscillators’ means and standard deviations, with temporal averaging and complex conjugation for complex series.
- For static chimeras, a nonzero distribution near |ρ| ≈ 1 indicates that the coherent cluster remains localized over time.Oscillators are treated as correlated when |ρ| > 0.99 = γ.
- The Kuramoto correlation distribution has a distinct peak at |ρ| = 1, identifying the chimera as static because most oscillators retain their group affiliation.A secondary peak near |ρ| ≈ 0.5 reflects partial linear correlation between oscillators near x ≈ 0.5 and synchronous oscillators.
- For the Kuramoto model, the time-correlated fraction is approximately 0.28.The reported value is given as 0.08 ≈ 0.28 in the passage describing the comparison with figure 5b.
- h0 does not always reflect synchronized-cluster size when coherent and incoherent regimes move spatially over time.It can become much smaller than g0(t), or vanish over sufficiently large time windows; equality occurs only for static chimeras without spatial coherence in the incoherent cluster.
III. EXAMPLES OF CHIMERA STATES AND THEIR CHARACTERIZATION
The proposed measures distinguish stationary, breathing, turbulent, and non-chimera dynamics across numerical and experimental examples. Their behavior in time and space reveals whether coherent and incoherent regions persist, oscillate, or disappear.
- Stationary chimeras have time-independent coherent and incoherent patches, with finite g0(t) and h0 indicating spatial and temporal incoherence.
- Static period-2 chimeras have constant g0(t) between 0 and 1 while h0 = 1, indicating spatial coexistence without temporal incoherence.
- Breathing type II chimeras show oscillatory g0(t), reflecting partial synchronization in the incoherent regime and homogeneous patches within the incoherent cluster.
- Turbulent type I chimeras exhibit irregular g0(t) while significantly positive h0 indicates static coherent and incoherent regions; irregularity arises from spatiotemporal intermittency.
- Spatiotemporal intermittency is not classified as a chimera because g0(t) repeatedly drops to zero, while h0 remains small from neighboring-point correlations.
- Localized turbulence contains erratically moving bubble-like turbulent islands, making h0 small and eventually vanishing over sufficiently long time windows.
A. Transient chimeras
The paper distinguishes transient chimeras from states that remain attractive in the continuum limit. Examples include amplitude chimeras and dynamics whose partial synchronization declines until full synchronization or incoherence is reached.
- A. Transient chimeras: Long-term stability matters because some chimeras are attractors, whereas others are long-term transients or collapse to a homogeneous state.
- A. Transient chimeras: Amplitude chimeras contain two coherent antiphase domains separated by a spatially incoherent interface, but transition to full synchronization after finite time.
- A. Transient chimeras: Transient chimeras are defined by 0 < g0(t) < 1 before a transient time t0, followed by g0(t) = 1 or g0(t) = 0.
- A. Transient chimeras: In the CO-oxidation model, declining g0(t) eventually vanishes, mediating a transition from an unstable to a stable state and satisfying the transient-chimera criteria.
B. Experimental observation of chimeras
The experimental data show a small coherent region within an inhomogeneously oscillating background. The measures g0(t) and h0 classify this observed experimental chimera as breathing.
- Figure 16a shows strong linear correlations within the coherent cluster that diminish toward the remaining oscillators.
- The temporal behavior of g0(t) and the value of h0 resemble the type II dynamics shown in figure 8.
- The experimental chimera is classified as breathing, while the small h0 reflects the relatively small coherent cluster.
IV. CLASSIFICATION SCHEME
The scheme uses g0(t) and h0 to quantify spatial and temporal coherence, define chimera states, and classify them across simulated and experimental data. It distinguishes stationary, turbulent, and breathing behavior, then subdivides these classes by temporal coherence while retaining explicit scope limits.
- g0(t) measures total coherent and incoherent proportions, whereas h0 captures temporal coherence but not local structure within the incoherent group.
- The classification scheme is summarized in figure 17, which assigns the examples discussed in the paper to its categories.
- The method applies to any dynamical system rather than only phase oscillators, enabling classification of general chimera states.
- States with 0 < g0(t) < 1 satisfy the proposed coexistence criterion and are classified as chimera states.
- Stationary chimeras have constant g0(t), turbulent chimeras have irregular g0(t), and breathing chimeras have periodic g0(t).
- Assignment can be ambiguous because the boundaries between stationary and turbulent, and turbulent and oscillatory, are fluid.
- The three groups can be further divided into subclasses using the temporal correlation measure h0.
- The measures discriminate different chimera types in simulated or experimental spatio-temporal data, and the approach was tested with FitzHugh-Nagumo and Rössler models.