Higher-Order Phase Reduction

• Higher-order phase reduction is a mathematical method that reduces complex oscillator dynamics with multi-body interactions to effective phase equations, preserving higher-order structures.
• It systematically extends classical phase reduction by rigorously capturing hypergraph and simplicial complex topologies, enabling analysis of synchronization and multistability.
• Applications include predicting cluster formation and chaotic regimes in oscillator networks, as demonstrated in Stuart-Landau systems with both pairwise and three-body interactions.

Higher-order phase reduction is a mathematical methodology for reducing the dynamics of coupled nonlinear oscillators with many-body interactions to effective equations governing only their phases, systematically extending classical phase reduction (which is typically limited to pairwise interactions and first-order coupling). This approach is crucial for understanding and analyzing synchronization, cluster formation, and multistable states in systems where the interaction structure transcends simple pairwise connectivity, encompassing hypergraph or simplicial complex topologies. Recent theory rigorously establishes how higher-order topologies, coupling symmetries, and interaction functions are preserved and manifested in the reduced phase models, thereby providing a unified analytical route to higher-order Kuramoto-type models (León et al., 27 Mar 2025, Battiston et al., 6 Oct 2025).

1. Mathematical Foundations and Generalization to Higher-order Interactions

Classical phase reduction reduces the dynamics of multidimensional limit-cycle oscillators to a single phase coordinate $\theta_j$, particularly when coupling between oscillators is weak. For pairwise coupled systems: $$\dot{\theta}_j = \omega_j + \epsilon \sum_k A_{jk} \Gamma_{jk}(\theta_k - \theta_j)$$ where $A_{jk}$ encodes network topology, and $\Gamma_{jk}$ is the phase coupling function, typically calculated from the oscillator's phase response curve.

Higher-order phase reduction generalizes this construction to account for interactions involving more than two oscillators. The general dynamical system for three-body (or in general, $q$-body) coupling is represented as: $$\dot{\bm{X}}_j = \bm{F}_j(\bm{X}_j) + \epsilon \sum_{k,l} A_{jkl} \bm{g}_{jkl}(\bm{X}_k, \bm{X}_l, \bm{X}_j)$$ where $A_{jkl}$ is a rank-3 adjacency tensor specifying hypergraph topology and $\bm{g}_{jkl}$ is a general interaction term.

Phase reduction up to first-order in $\epsilon$ yields: $$\boxed{ \dot{\theta}_j = \omega_j + \epsilon \sum_{k,l} A_{jkl} \Gamma_{jkl}(\Delta\theta_{kj}, \Delta\theta_{lj}) + O(\epsilon^2) }$$ where $\Delta\theta_{kj} = \theta_k - \theta_j$ and

$$\Gamma_{jkl}(\Delta\theta_{kj}, \Delta\theta_{lj}) = \frac{1}{2\pi} \int_0^{2\pi} \bm{Z}(\varphi) \cdot \bm{p}_{jkl}(\Delta\theta_{kj} + \varphi, \Delta\theta_{lj} + \varphi, \varphi) d\varphi$$

with $\bm{Z}(\varphi)$ the phase sensitivity function. This formalism extends naturally to arbitrary interaction order and mixtures of pairwise and higher-order terms.

2. Preservation of Higher-order Topology and Hypergraph Structure

A central theoretical advance is the demonstration that phase reduction does not merely encode the interaction strength—it preserves the full incidence and structure of the hypergraph or simplicial complex underlying the physical coupling (León et al., 27 Mar 2025). The adjacency tensor $A_{jkl\ldots}$ in the original model directly maps onto the adjacency tensor in the reduced phase equations up to first order in $\epsilon$, meaning that higher-order topological features (such as specific groupings in hyperedges, or faces of a simplicial complex) are exact in the corresponding phase model. Analytical investigation reveals that phase models defined on hypergraphs or higher-order networks can thus be rigorously derived from complex oscillator dynamics, enabling direct analysis of synchronization, cluster states, and multistable regimes inherent to systems with nonpairwise interactions (Battiston et al., 6 Oct 2025).

3. Symmetry Constraints and the Role of Odd/Even Couplings

The effect of symmetry, such as rotational or antisymmetric properties of the oscillator vector field, is profoundly constraining in higher-order phase reduction (León et al., 27 Mar 2025). If the underlying oscillator exhibits antisymmetry (i.e., the limit cycle and phase response are odd functions), only the odd part of the coupling function participates in the leading-order phase dynamics. Even coupling terms, such as symmetric quadratic or quartic interactions, are rigorously shown to vanish at order $O(\epsilon)$ via Fourier expansion and averaging procedures. This analytic result accounts for the observed prominence of odd (e.g., cubic) couplings in both natural and engineered oscillator networks and justifies model reductions that disregard even high-order coupling under appropriate symmetry assumptions.

4. Explicit Applications: Stuart-Landau Oscillator Networks

Stuart-Landau oscillators provide analytically tractable testbeds for higher-order phase reduction due to their intrinsic rotational symmetry.

• All-to-all configuration:

For globally coupled populations with both pairwise ($K_1$) and three-body ($K_2$) interactions:

$$\dot{\theta}_j = \omega + \frac{K_1 R}{2} \sin(\Psi - \theta_j) + \frac{K_2 R^2}{8} \sin(2\Psi - 2\theta_j) + \text{const.}$$

with $R e^{i\Psi} = \frac{1}{N} \sum_k e^{i\theta_k}$. The phase model admits analytical prediction of synchronization onset, cluster formation, and multistability, closely mirroring direct simulation outcomes.

• Ring-like hypergraph topology:

For a network where each oscillator interacts in three-body hyperedges with its nearest neighbors (hyperring), the reduced phase equation is:

$$\dot{\theta}_j = \omega + \frac{K_1}{2} [\sin(\theta_{j-1} - \theta_j) + \sin(\theta_{j+1} - \theta_j)] + \frac{K_2}{8} \sin(\theta_{j+1} + \theta_{j-1} - 2\theta_j)$$

This formulation captures phenomena such as twisted states and their stability regimes, confirmed by simulation, and enables direct phase diagram construction.

5. Higher-order Kuramoto Models and Analytical Tractability

The generalized theory reveals that higher-order Kuramoto models, including classical pairwise, three-body, and more general multi-body sine-coupling phase equations, systematically arise from nonlinear oscillator networks via phase reduction (León et al., 27 Mar 2025, Battiston et al., 6 Oct 2025). The reduced equations, retaining the topological structure and symmetry constraints, allow for tractable bifurcation and stability analysis of synchronized, clustered, and chaotic states. For example, the higher-order Kuramoto model can take forms such as: $$\dot{\theta}_j = \omega + \epsilon \sum_{k,l} A_{jkl} [ \kappa_1 \sin(\theta_k + \theta_l - 2\theta_j + \alpha) + \kappa_2 \sin(\theta_k - \theta_l + \beta) ]$$ or more generally as

$$\dot{\theta}_k = \omega + \sum_j g_2(\theta_j-\theta_k) + \sum_{j,l} g_3(\theta_j+\theta_l-2\theta_k) + \ldots$$

where the functions $g_q$ encode phase interactions of order $q$. The theory enables analytical treatment of explosive synchronization, multistability, cluster states, and chaos attributed entirely to higher-order coupling structure.

6. Impact, Open Questions, and Future Directions

Higher-order phase reduction is foundational for the analysis of collective dynamics in physical, biological, or technological networks with nonpairwise connectivity. Key impacts include:

• Emergence and analysis of new dynamical regimes: Explosive transitions, multistability, and chaotic attractors can be classified and predicted via the structure of many-body phase interactions.
• Justification and systematic derivation of higher-order phase oscillator models: The proliferation of Kuramoto-like models with three-body (or higher-body) sine coupling is shown to be a rigorous outcome of phase reduction in complex networks.
• Guidance for network design and control: Insights into which coupling symmetries and topologies lead to tractable dynamics or desired collective states.

Open questions include the characterization of optimal coupling functions for synchronization, dimensionality reduction methods for high-order interaction Laplacians, distinguishing physically present from emergent effective interactions via phase reduction, and inference methods for reconstructing hypergraph structure from observed dynamics (Battiston et al., 6 Oct 2025).

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Feature	Pairwise Reduction	Higher-order Phase Reduction
Topology captured	Simple graphs (adjacency)	Hypergraphs, simplicial complexes
Leading-order interaction	Pairwise sine, PRC forms	Multi-body coupling; odd terms dominant
Symmetry constraints	Pairwise antisymmetry effects	Odd couplings survive; even vanish
Dynamical regime	Synchrony, incoherence	Multistability, cluster, chaos, explosive
Analytical tractability	Classic Kuramoto analysis	Extended Kuramoto models, bifurcations

• Theory of phase reduction from hypergraphs to simplicial complexes: a general route to higher-order Kuramoto models (León et al., 27 Mar 2025)
• Collective dynamics on higher-order networks (Battiston et al., 6 Oct 2025)