Amplitude Chimeras in Coupled Oscillators

• Amplitude chimeras are spatiotemporal patterns in oscillator networks defined by coexisting domains of coherent and irregular amplitudes amid global phase synchronization.
• They emerge from nonlocal or symmetry-breaking coupling, bistable local dynamics, and auxiliary coupling layers that drive multistability and spatial segregation.
• Diagnostic measures like amplitude center of mass and mean phase velocity profiles provide insights into their stability and transition regimes.

Amplitude chimeras are spatiotemporal patterns in coupled oscillator networks distinguished by the coexistence of spatially coherent and incoherent groups with respect to oscillation amplitude, while phase or frequency synchronization is typically preserved network-wide. These states arise primarily in networks where nonlocal or symmetry-breaking coupling, bistable local dynamics, or auxiliary @@@@1@@@@ are present. A defining trait is the partition of the system into domains in which oscillator amplitudes either synchronize (coherently oscillate with common properties) or remain spatially irregular (neighboring oscillators exhibit uncorrelated amplitude levels), despite uniform system parameters and symmetric interactions.

1. Mathematical Characterization and Model Classes

Amplitude chimeras have been identified across a range of dynamical systems. The prototypical models include:

• Nonlocally coupled Stuart-Landau (SL) oscillator rings:

$$\dot z_j = (1 + i\omega - |z_j|^2)z_j + \frac{\sigma}{2P} \sum_{k=j-P}^{j+P} (\Re z_k - \Re z_j)$$

where $\omega$ is the natural frequency, $\sigma$ the coupling strength, and $P$ the coupling range. Here, coupling acts only through the real part to break continuous $S^1$-symmetry (Zakharova et al., 2015Tumash et al., 2016).

• Generic oscillator networks with nonlocal, environment-mediated, or delayed coupling, e.g., two-layer networks where one layer of local oscillators is coupled to a second "environment layer" with nonlocal interactions (Verma et al., 2020), or inclusion of time-delay in network links (Gjurchinovski et al., 2017).
• Bistable/chaotic maps and reaction-diffusion systems, where local multistability induces amplitude domains (e.g., cubic map lattices, Van der Pol or prey–predator PDEs) (Shepelev et al., 2016Kaper et al., 2021Kundu et al., 2021).

Amplitude chimeras differ fundamentally from phase chimeras: in amplitude chimeras the phase velocities remain constant across the network but amplitude order is broken, while in phase chimeras the amplitude is uniform and phase synchronization is lost only in incoherent domains (Banerjee et al., 2018, Loos et al., 2015).

2. Dynamical Mechanisms and Formation Scenarios

Amplitude chimera states typically originate via bifurcation routes involving symmetry-breaking or multistability:

• Symmetry-breaking and bifurcation routes: In SL or Rayleigh oscillator networks, pitchfork bifurcations of inhomogeneous steady states can produce intervals where unstable periodic orbits fill the phase space, and nonlocality further supports domain formation (Banerjee et al., 2018Zakharova et al., 2015). The AC state occupies a "trapped region" densely populated by such orbits, leading to long-lived transients.
• Environment-mediated reinforcement: In multilayer models, a local oscillator network (Layer 1) is coupled (via amplitude channel) to a second network (Layer 2) with nonlocal interactions. Inhomogeneities in Layer 2 reinforce spatial amplitude inhomogeneity in Layer 1, stabilizing amplitude chimeras even if Layer 1's native topology is purely local (Verma et al., 2020).
• Driven mean-field and global coupling: In globally coupled systems containing active and inactive oscillators, mean-field drives create bistable conditions where amplitude-mediated chimeras form as some oscillators settle into large-amplitude limit cycles, others into lower-amplitude or fixed-point states (Mukherjee et al., 2017, Sethia et al., 2013).
• Bistable or multi-attractor local dynamics: In coupled cubic maps or Van der Pol/FHN systems with multiple local attractors, nonlocal coupling leads to coexistence of domains at different amplitude levels. Interfaces act as incoherent regions (Provata, 2023Shepelev et al., 2016Kaper et al., 2021).

3. Classification, Diagnostics, and Statistics

Amplitude chimera identification involves both spatial and temporal observables:

• Amplitude center of mass: For oscillator $j$, the period-averaged center,

$$y_{\mathrm{cm},j} = \frac{1}{T} \int_0^T y_j(t)\,dt$$

is zero in coherent domains, nonzero and spatially irregular in incoherent ones (Zakharova et al., 2015Banerjee et al., 2018Premalatha et al., 2018).

• Mean phase velocity profile: Remains flat across coherent and incoherent domains, distinguishing amplitude chimeras from phase chimeras (Zakharova et al., 2015Banerjee et al., 2018).
• Strength of incoherence measure:

$$S = 1 - \frac{1}{M} \sum_{m=1}^M \Theta(\delta - \sigma_m)$$

where local standard deviations $\sigma_m$ are computed over spatial bins; $0Zakharova et al., 2015Verma et al., 2020).

• Global order parameters: Higher-order Kuramoto-type order parameters, $R^n$, are used to detect collapse of amplitude chimeras to coherent waves or traveling waves (Gjurchinovski et al., 2017).
• Lifetime statistics: Transient amplitude chimeras exhibit lifetimes that decrease with system size ($t_{\mathrm{tr}} \sim N^{-2} + \text{const}$) and logarithmically with noise intensity ($t_{\mathrm{tr}}(D) \approx -\ln D/\mu + \eta$), saturating at a finite limit as $N\to\infty$, with sensitivity to initial-symmetry properties (Loos et al., 2015).

4. Stability, Control, and Transition Regimes

Amplitude chimeras appear as dynamically metastable or even genuinely stable states depending on system parameters and architecture:

• Stability via Floquet analysis: In typical SL networks, amplitude chimeras are classified as high-dimensional saddle states, exhibiting at least one positive Floquet exponent corresponding to unstable manifolds in phase space. Their lifetimes are inversely proportional to the largest positive exponent (Tumash et al., 2016Premalatha et al., 2018). In environment-mediated or reinforced settings, the periodic solution can achieve linear stability ($|\mu_{i>1}|<1$ for all non-phase modes) (Verma et al., 2020).
• Regime diagrams and parameter dependence:
  • For nonlocally coupled oscillator rings, amplitude chimeras form in an intermediate window of coupling strength and range; outside this window, the system organizes into full synchrony, traveling waves, oscillation death, or chimera-death (CD) states (Zakharova et al., 2015Verma et al., 2020).
  • In the two-layer setting (Verma et al., 2020),
  • Small interlayer coupling $\epsilon \lt 1$: complete synchronization
  • Intermediate $\epsilon$ and nonlocality $R_2 > 0.1$: stable amplitude chimeras
  • Large $\epsilon \gtrsim 6$: chimera death and multicluster steady states.
• Control strategies: Time delay in coupling (Gjurchinovski et al., 2017), tunable nonlocality of coupling or environmental layer (Verma et al., 2020), and network topology modifications (e.g., star-ring amelioration (Muni et al., 2020)) allow for deliberate suppression, enhancement, or stabilization of amplitude chimeras.
• Breathing and composite chimeras: Bifurcation scenarios such as saddle-node and Hopf bifurcations lead to breathing amplitude chimeras (oscillatory spatial patterns in amplitude), multicluster chimeras, and, when amplitude and frequency are entangled, composite amplitude-frequency chimeras (Provata, 2023Banerjee et al., 2018Gjurchinovski et al., 2017).

5. Variants, Generalizations, and Related Phenomena

Multiple amplitude chimera variants exist, and related concepts have emerged:

• Amplitude-mediated chimeras (AMC): In CGL/NLCGL equations, AMCs bridge pure phase chimeras and regime turbulence, often featuring amplitude "holes" in incoherent domains (1304.53821503.04053).
• Chimera death (CD): With sufficient coupling, amplitude chimeras can transition directly to inhomogeneous steady states, combining features of oscillation death and spatial chimera patterns: spatially coherent and incoherent static domains (Zakharova et al., 2015Verma et al., 2020).
• Spatiotemporal and spiral chimeras: Amplitude chimeras can be embedded in rotating spiral patterns in reaction-diffusion systems, typically forming incoherent spiral cores and coherent arms (Kundu et al., 2021).
• Small and localized chimeras: Even in small networks (as few as $N=3$), chimera-like amplitude desynchronization arises, with multistable attractor basins underpinned by transient chaos (Banerjee et al., 2018).
• Mixed-amplitude and variable amplitude chimeras: Patterns exhibiting coexisting large-amplitude, low-frequency and small-amplitude, high-frequency domains, characterized by local dynamics (e.g., canard-induced MMOs vs SAOs) and parameter-dependence of coexistence windows (Kaper et al., 2021).
• Bump states: Amplitude chimeras continuously transform into bump states where inactive (zero-amplitude) and active domains coexist as amplitude attractors merge into attractors or collapse (Provata, 2023).

6. Experimental and Theoretical Applications

Amplitude chimeras have direct theoretical relevance and potential physical realizations:

• Laser arrays and coherent photonics: Chimera states involving amplitude, phase, and inversion have been realized in globally coupled class-B semiconductor laser arrays, requiring non-negligible amplitude-phase coupling and multistability near steady-state bifurcation boundaries (Böhm et al., 2014).
• Chemical and electrochemical oscillators: Nonlocal mean-field or global-feedback reactions enable AMC states, particularly in spatially extended reactors or oscillatory media (1304.53821503.04053).
• Neuroscience and biological networks: Spatiotemporal amplitude chimeras model partial synchrony in neural systems, including sleep-state hemispheric asymmetry and patterns of neuronal activity with age- or damage-driven inactive subpopulations ("robustness to damage" via mean-field coupling has been analyzed (Mukherjee et al., 2017)).
• Complex technical and engineered networks: Superconducting quantum devices (SQUID arrays), Chua circuit networks, and genetically engineered oscillatory systems present platforms for engineered amplitude chimera control and study (Banerjee et al., 2018Muni et al., 2020).
• Generic mechanism across models: The generic mechanism of amplitude chimeras involves the coexistence of multiple attractors, spatially structured coupling, and bifurcation sequences allowing stabilization or long-lived transients.

7. Outlook and Theoretical Significance

Amplitude chimeras have established themselves as a robust and universal manifestation of partial coherence in high-dimensional dynamical systems with multistable or symmetry-broken interactions. Their formation is closely tied to the interplay of spatial coupling (especially nonlocality), intrinsic oscillator multistability, and environmental feedback. Theoretical models demonstrate that amplitude chimeras are not limited to ideal ring or regular networks but extend to multilayer, complex topology, and even systems with strong heterogeneity or sparse connectivity.

Their persistence (transient or stable), sensitivity to system size, initial conditions, and noise, and amenability to control by local/global/topological parameters highlight the flexibility and diversity of this dynamical regime. Understanding amplitude chimeras opens avenues in designing and controlling collective dynamics in distributed systems, from technological oscillator arrays to biological tissues and beyond (Verma et al., 2020Zakharova et al., 2015Premalatha et al., 2018Loos et al., 2015Gjurchinovski et al., 2017).