Atypical Chimera States

• Atypical chimera states are dynamical regimes in coupled oscillators that defy classical symmetry, topology, or order-parameter constraints, showing mixed coherent and incoherent behavior.
• They emerge from the interplay of network topology, nonlocal and higher-order interactions, and nonlinear dynamics, as evidenced in spiral-wave, directed, and amplitude-mediated models.
• Mathematical models utilizing phase, amplitude, and high-order coupling provide diagnostic tools and clarify transitions, offering insights for applications in neuroscience, power grids, and complex systems.

Atypical chimera states are dynamical regimes in coupled oscillator networks that exhibit coexistence of coherent and incoherent subpopulations, but violate one or more defining properties of classical chimeras, such as topology, symmetry, order-parameter structure, or mechanisms of incoherence. These states have been rigorously identified in a variety of systems including two-dimensional oscillator lattices with nonlocal coupling, directed and higher-order networks, mixed coupling schemes, hybrid continuous–discrete settings, and even minimalistic cellular automata. Their emergence is governed by nontrivial interplay between network topology, coupling symmetry, nonlinear interaction terms, and bifurcation structure, revealing a much richer taxonomy than initially recognized in standard nonlocally coupled phase oscillator rings.

1. Conceptual Definition and Expanded Taxonomy

The canonical chimera state, as defined in the early work of Kuramoto, Battogtokh, Abrams, and Strogatz, features a nonlocally coupled one-dimensional ring network of identical phase oscillators that splits spontaneously into a spatially contiguous domain of phase-locked oscillators and a complementary incoherent (drifting) domain. Atypical chimera states are those that violate at least one of these classical constraints: they may occur in non-spatial or higher-dimensional topologies, exhibit coherent/incoherent partitioning in index rather than geometric space, involve non-phase order parameters, lack strict frequency separation between domains, or arise by mechanisms other than homoclinic or SNIPER bifurcation of a global phase variable (Haugland, 2021).

Prominent atypical classes include:

• Defect-mediated chimeras with topologically sustained incoherence (spiral cores, vortex nucleation) (Liu et al., 26 Nov 2025),
• Amplitude-mediated chimeras lacking clear phase/frequency separation (Sethia et al., 2013),
• Hybrid or layered coherence-incoherence (e.g., coherent–incoherent–twisted states, anti-phase clusters) (Shcherbakov et al., 30 Nov 2025, Gopal et al., 2017),
• Chimeras arising under purely local, global, or high-order (hypergraph) interactions (Djeudjo et al., 14 Jun 2025, Meena et al., 2015, García-Morales, 2016),
• Symmetry-broken states in small networks not reducible to spatial contiguity (Kemeth et al., 2018).

The recognition of these states has prompted significant re-evaluation of what constitutes a chimera and led to systematic classification efforts using generalized order parameters, symmetry diagnostics, and bifurcation-theoretic analysis (Haugland, 2021).

2. Mathematical Models and Governing Equations

Atypical chimeras are instantiated in diverse dynamical systems, each revealing new organizing principles beyond classical nonlocal phase coupling. Notable examples include:

• 2D Nonlocally Coupled Oscillator Lattice (Spiral-Wave Chimera):

$$\frac{d\phi(\bm r)}{d\tau} = -\,\frac{1}{N(R)}\sum_{\bm r':0<|\bm r-\bm r'|\le R} \sin\bigl[\phi(\bm r)-\phi(\bm r')+\alpha\bigr]$$

with topological defect detection via discrete and continuum winding numbers (Liu et al., 26 Nov 2025).

• Directed Networks:

$$m\,\ddot{\theta}_i + \varepsilon\,\dot{\theta}_i = \omega + \frac{\mu}{P}\sum_{j=1}^{P} \sin(\theta_{i+j}-\theta_i-\alpha)$$

where strictly one-way coupling generates robust solitary and multi-solitary chimeras over extended parameter domains (Jaros et al., 2021).

• Higher-Order m-Directed Hypergraphs:

$$\dot\theta_i = \omega + \varepsilon\!\sum_{j_1, j_2} A^{(2)}_{i, j_1, j_2}\,\Phi(\theta_i, \theta_{j_1}, \theta_{j_2})$$

with non-reciprocal, many-body interaction structure enabling traveling and persistent chimeras not found in pairwise-directed networks (Djeudjo et al., 14 Jun 2025).

• Hybrids of Continuous and Discrete Architectures:

Minimalistic cellular automata with majority-voting and chaotic sublayers demonstrate spatially static, robust chimera patterns parameterized by nonlocal coupling range, where incoherent and coherent domains emerge without any reference to amplitude/phase variables (García-Morales, 2016).

• Mixed-Mode and Environmental Coupling:

Ensembles subject to phase-flip bifurcations via dynamic environmental fields produce coherent domains phase-locked at relative π shift, with interspersed incoherent regions (Gopal et al., 2017).

3. Mechanisms of Emergence and Distinction from Classical Chimeras

Atypical chimeras arise from mechanisms that are generically distinct from the standard phase-balance instability seen in the Kuramoto–Battogtokh or one-dimensional ring models. Key examples:

• Geometry–Topology Duality (Spiral-Wave Chimera):

In the small phase-lag regime, incoherent core size grows linearly with α (geometric effect), but as α increases past a threshold, core expansion gives way to an exponential proliferation of topological defects (vortex nucleation/annihilation), with statistical signatures reminiscent of a Berezinskii–Kosterlitz–Thouless (BKT) transition (Liu et al., 26 Nov 2025).

• Dynamic Symmetry Breaking (Directed/Hybrid Networks):

In directed oscillator systems, asymmetry-induced heteroclinic interactions and saddle-node bifurcations of limit-cycle orbits produce robust islands and continents of chimera behavior, with solitary/incoherent groups determined by the interplay of directionality and cyclic feedback (Jaros et al., 2021).

• High-Order Interaction Effects (Hypergraphs):

Many-body, non-reciprocal coupling structures on m-directed hypergraphs restore coexistence of incoherence and coherence lost in directed pairwise networks, with block structures determined by effective buffering and nontrivial amplitude-phase coupling (Djeudjo et al., 14 Jun 2025).

• Motion-Induced Kernel Asymmetry (Mobile Particle Ensembles):

Nonuniform, periodic spatial motion of oscillators modulates the effective (nonlocal) coupling kernel and can bifurcate classical chimeras into twisted or alternating coherent–incoherent–twisted composite domains (Shcherbakov et al., 30 Nov 2025).

• Amplitude Mediation and Global Coupling:

Strong mean-field coupling and amplitude dynamics (as in the globally coupled complex Ginzburg–Landau equation) yield chimeras not associated with phase drift, but rather by coexistence of amplitude-locked and amplitude-modulated oscillator subgroups, outside the classical weak-coupling, phase-only context (Sethia et al., 2013, Haugland, 2021).

4. Statistical and Topological Diagnostics

Detecting and characterizing atypical chimeras requires metrics adapted to their nonconventional architecture. Approaches include:

• Winding Number Statistics: Topological charge density (vortex count) and its scaling laws with phase-lag reveal transitions between defect-free and defect-proliferating chimera regimes, with transitions from binomial to Poissonian statistics at critical α∗ (Liu et al., 26 Nov 2025).
• Order Parameters Beyond Phase: Amplitude, envelope, and frequency-resolved order parameters uncover regimes like amplitude-mediated chimeras, “chaotic amplitude chimeras,” and states where incoherent oscillators remain frequency-locked but desynchronize in amplitude/trajectory (Calim et al., 2020, Shcherbakov et al., 30 Nov 2025).
• Local Cross-Correlation Measures: Space-resolved cross-correlation and firing-rate heterogeneity detect both conventional and hybrid twisted/incoherent domains, with the ability to sharply distinguish spatially correlated clusters versus genuinely incoherent subpopulations.
• Symmetry Detectives: In small networks, set-wise and instantaneous symmetry measures (via permutation group action) classify chimeras by presence/absence of group invariances, distinguishing classes with and without statistical interchangeability among incoherent units (Kemeth et al., 2018).
• Bifurcation and Basin Stability Analysis: For nonlinear network models, transitions between classical and atypical chimera states, as well as multistability and robustness, are mapped using bifurcation structure and basin stability estimators (Meena et al., 2015, Gopal et al., 2017).

5. Underlying Topology, Structural Robustness, and Experimental Realizability

Atypical chimera states are fundamentally shaped by the topology of coupling and the symmetry/breaking thereof:

Chimera Class	Coupling Topology	Key Robustness Features
Spiral-wave chimera	2D nonlocal lattice	BKT-like defect transitions, dual scaling
Star-network chimera	Star (hub-and-spoke)	Wide parameter windows, high basin stability
Hypergraph-induced chimera	m-directed hypergraph	Persistence under non-reciprocity and many-body interactions
Mobile oscillator chimera	1D ring, time-dependent	Kernel-induced broken symmetry, alternation
CA chimera	Discrete CA ring	Static coherent/incoherent domains, minimal parameter set
Amplitude-mediated chimera	Mean-field (global)	Stable in strong coupling and inhomogeneous amplitude regimes

Many of these regimes have been validated in experimental analog/bioelectrical settings (e.g., star-network Rössler circuits, brain-inspired hybrid neuron networks), have rich parameter windows (star topology, hypergraphs), and can arise robustly from generic random initial conditions, in contrast with the parameter fragility and IC-dependence often seen in classic 1D nonlocal chimeras (Meena et al., 2015, Haugland, 2021).

6. Broader Implications and Classification Criteria

The recognition of atypical chimeras has spurred the development of generalized taxonomies and classification frameworks. Criteria now include:

• Amplitude mediation versus pure phase drift,
• Homogenization of long-term frequencies among all oscillators,
• Existence under non-spatial, local, or global coupling,
• Discrete-time or cellular automaton realizability,
• Lack of strict spatial contiguity or topological-geometric correlation of coherent/incoherent domains,
• Statistical transitions (e.g., binomial to Poisson) signaling topological phase transitions.

This broadening of the chimera concept has relevant implications for neuroscience (e.g., unihemispheric sleep, working-memory “bump” states), power-grid and optical networks, and the general theory of spontaneously symmetry-broken collective dynamics. The expanded variety of mechanisms and topologies that support chimera behavior suggests that partial coherence–incoherence coexistence is a pervasive feature of complex dynamical systems, demanding careful theoretical, numerical, and experimental analysis for correct identification and practical utilization (Haugland, 2021).