Singular Basins in Adaptive Oscillator Networks

• Singular Basins are regions in phase space of adaptive oscillator networks characterized by abrupt boundaries and emergent behaviors such as metastability and clustering.
• They arise from the interplay of nonlinear feedback, slow–fast dynamic coupling, and network topology, leading to complex bifurcation phenomena.
• Understanding singular basins enhances insights into synchronization thresholds and modular dynamics in systems ranging from neuroscience to engineered oscillators.

A singular basin, in the context of adaptive and complex oscillator networks, refers to a set of initial conditions in phase–space leading to a particular asymptotic collective state, where the basin's geometrical or dynamical properties exhibit non-regular structure, abrupt boundaries, or are associated with emergent behaviors such as metastability, cluster formation, or collective excitability. Singular basins—often distinguished from simple attractor basins—are fundamentally shaped by adaptation mechanisms, network topology, multistability, and slow–fast dynamic interplay.

1. Mathematical Definition and Context

Singular basins arise in systems where the attractors governing long-term dynamics possess complex or non-smooth basin boundaries, often due to the structure of adaptive coupling and nonlinear feedback. In adaptive phase-rotator networks, the state space comprises the phases $\theta_n(t)$ of each node, along with time-dependent coupling variables (such as $S_n(t)$, $\kappa_{ij}(t)$), whose evolution follows distinct slow–fast timescales. The basins are defined by the partition of initial conditions leading to distinct collective behaviors; for example, synchronization, cluster formation, solitary states, or excitable events. In singular settings, these basin boundaries may shift abruptly due to bifurcations induced by adaptation, resulting in sharp thresholds in response to perturbations (Paolini et al., 2022, Berner et al., 2019).

2. Emergence in Adaptive Oscillator Networks

Adaptive networks of phase rotators exemplify singular basin phenomena through nontrivial bifurcation structures. The feedback adaptation mechanisms—such as the linear feedback $S_n(t)$ targeting $K-\alpha Q_n(t)$—drive the system across a hysteretic phase transition, generating collective states whose attraction basins can be singular, i.e., characterized by abrupt “walls” between different dynamical responses. The adaptive Kuramoto–Sakaguchi models, particularly under slow adaptation, demonstrate multistable regimes (e.g., anti-phase clusters, in-phase synchronous states, partial coherence) with basin boundaries determined by critical coupling, phase-lags, and learning rules (Jüttner et al., 2022, Berner et al., 2019).

3. Multistability and Basin Geometry

The formation of hierarchical frequency clusters and bistable states yields basins with geometric singularities, such as codimension-one stable/unstable manifolds at saddle equilibria. For instance, adaptive Kuramoto models exhibit antipodal, splay, and double-antipodal clusters, with fixed points and limit cycles defined by

$$\epsilon < \epsilon_c = -\frac14\sin(\alpha-\beta) + \frac12 \sqrt{\frac14\sin^2(\alpha-\beta) + (n_1-\frac12)^2\cos^2(\alpha-\beta)}$$

and basin boundaries partitioned by critical adaptation speed (Berner et al., 2019). Double-antipodal states—although linearly unstable—structure transient flow, generating metastable “singular basins” through heteroclinic orbits connecting antipodal to splay clusters. These geometrical features manifest in extended dwell times near saddle-type attractors, indicative of singularity in basin topology (Berner et al., 2019, Berner et al., 2019).

4. Excitability and Threshold Phenomena

Adaptive feedback across a hysteretic transition can produce singular basins associated with collective excitability. Networks of non-excitable rotators display all-or-none macroscopic responses: an external perturbation must exceed a well-defined threshold amplitude $A_0$ or stimulated fraction $P_c$ to “kick” the system from one basin to another—characterized by power-law scaling near activation thresholds: $$A_0(M) - A_0(\infty) \propto M^{-\beta}, \quad R_m(\infty) - R_m(M) \propto M^{-1}$$ Basin boundaries demarcate phase transition-like phenomena, separating no-response and excitable burst regimes, and remain sharply defined even under extensive network dilution (Paolini et al., 2022).

5. Solitary and Cluster States: Singular Basin Manifestations

Singular basins are also evident in the stability and onset of solitary states and multicluster configurations. In nonlocal adaptive rings, solitary or low-population clusters bifurcate via pitchfork and homoclinic mechanisms, with basins shrinking or expanding dramatically as adaptation parameters ($\alpha,\beta,\epsilon$) vary (Berner et al., 2019). For multicluster solutions, basin structure is organized by the interplay of adaptation strength and cluster sizes, leading to sudden appearance/disappearance of attractors—critical for understanding modular and metastable organization in real-world networks (Berner et al., 2019).

6. Singular Basins in Network Topology and Layer Structure

Multiplexing—a multi-layer topology—can induce and stabilize phase-cluster patterns with singular basin boundaries not present in single-layer networks. Analysis via Laplacian spectra and multiplex decomposition reveals that certain cluster states are “born-by-multiplex,” exhibiting new basins with abrupt stability domains as interlayer coupling strength crosses critical values (Berner et al., 2019). This mechanism is generic for partial synchronization patterns, lending singular geometry to basin boundaries across layers in adaptive networks.

7. Dynamical and Practical Implications

Singular basin phenomena elucidate how slow–fast adaptation, network topology, and nonlinearity interact to generate macroscopic order, metastability, and robustness in complex adaptive systems. They explain sharp collective response thresholds, clustering transitions, and the emergence of novel collective attractors—often absent in standard, nonadaptive models. Singular basin structure underpins critical phenomena relevant to synchronization in neuroscience, engineered oscillator arrays, and adaptive control protocols. These insights guide both theoretical understanding and practical design for robust collective dynamics in sparse, modular, and adaptable networks (Paolini et al., 2022, Berner et al., 2019, Berner et al., 2019, Jüttner et al., 2022).