Connecting Hodge and Sakaguchi-Kuramoto through a mathematical framework for coupled oscillators on simplicial complexes

Synchronization, a ubiquitous phenomenon across various complex systems, has been extensively studied using the Kuramoto model of coupled oscillators. Initially formulated with all-to-all coupling, the model has been generalized to inhomogeneous interactions represented by graphs, revealing how graph structure influences synchronization dynamics. Further variations, such as time delays, oriented or signed interactions, and stochasticity, have enriched the model's explanatory power. The introduction of a frustration parameter, as in the Sakaguchi-Kuramoto model, generates rich dynamics and finds application in diverse fields. Recently, the focus has shifted to higher-order interactions, which are often represented using hypergraphs or simplicial complexes, extending pairwise interactions to three-way, four-way, and higher interactions. Consequently, generalizations of known dynamical systems, including the Kuramoto model, have been proposed to study higher-order interactions. Two primary approaches exist for extending oscillator models: the first maintains node-based phases but upgrades interactions to polyadic form; the second promotes phase variables from nodes to higher-order simplices, coupled by boundary operators. This research builds upon the second approach, extending a previous simplicial Kuramoto model by incorporating weights on simplices and linear and nonlinear frustration terms. The challenge lies in introducing nonlinear frustration without orientation dependence; this is addressed by using lift matrices and projections to create an equivalent formulation independent of orientation. The study uses numerical simulations, Hodge decomposition, order parameters, and Lyapunov exponents to analyze the dynamics.

The Kuramoto model, a cornerstone in the study of synchronization, has seen numerous extensions to capture diverse synchronization behaviors. Early work focused on all-to-all coupled oscillators, but later generalizations considered inhomogeneous coupling represented by graphs. The influence of graph topology on synchronization dynamics has been a significant area of research, with studies showing how modular structures can be revealed through transient dynamics. Other extensions include incorporating time delays, oriented or signed interactions, and stochasticity. The Sakaguchi-Kuramoto model, which introduces frustration into the nonlinear term, has been shown to produce rich and complex dynamics, with applications ranging from Josephson junction arrays to power grids. The growing interest in higher-order interactions has led to the development of Kuramoto models on simplicial complexes, representing higher-order connections. Two main approaches have emerged: models with phases defined on nodes and higher-order interactions, and models with phases assigned to higher-order simplices themselves, coupled through boundary operators. Previous research exploring the second approach has highlighted explosive synchronization properties with specific non-local couplings. This paper directly builds upon this prior work by integrating weights and frustration terms into the model.

The paper's mathematical framework leverages discrete differential geometry, employing boundary operators and Hodge Laplacians. A k-simplex is defined by k+1 nodes (1-simplex is an edge, 2-simplex is a triangle, etc.), and a simplicial complex is a set of simplices where every face is also a simplex. k-chains are linear combinations of k-simplices, with coboundary operators (N_k and its dual N_k^†) defined using generalized incidence matrices (B_k^k+1) and weight matrices (W_k). The Hodge Laplacian (L_k) is defined as the sum of 'down' and 'up' Laplacians, incorporating both local and non-local interactions. The standard consensus formulation is used, but the framework is not limited to consensus dynamics. The weighted simplicial Kuramoto model is formulated as a time-dependent k-cochain θ^(k), extending the standard Kuramoto model to higher-order simplices. The introduction of frustration is achieved using lift matrices (V_k) which double the simplices with opposite orientations, and projection operators that select either positive or negative entries to maintain orientation invariance. The frustrated simplicial Kuramoto model incorporates linear and nonlinear frustration terms (a_k and a_k+1 respectively) through a frustration operator F^(k)(N_k). For k=1 (edge oscillators), the model reduces to a specific form which is independent of the orientation of faces, but not edges. The Hodge decomposition is applied to analyze the dynamics, decomposing the k-cochain into gradient, harmonic, and curl components. The simplicial order parameter (SOP) is introduced as a measure of synchronization, generalizing the standard order parameter to higher-order simplices. It quantifies the degree of phase locking, with a value of 1 indicating full synchronization within the harmonic space. Numerical simulations, employing several simplicial complexes (a single face, a complex with a hole, and a larger Delaunay triangulation) are used to analyze the dynamics under varying frustration parameters and edge orientations. The largest Lyapunov exponent is computed to quantify the chaotic nature of the dynamics.

The research reveals a variety of complex dynamical behaviors in the frustrated simplicial Kuramoto model. In a simple triangle complex, simulations reveal a full synchronization regime even with frustration, and the analysis using Hodge decomposition shows that the gradient component's non-constancy correlates with a drop in synchronization. A complex with a single hole displays dynamics sensitive to edge orientation. Changing edge orientation can dramatically impact the synchronization level and the behavior of gradient and curl components. The study identifies two critical values for the nonlinear frustration parameter (α₂): α₂^g, where the gradient becomes non-constant, and α₂^c, where the curl becomes non-constant (α₂^c ≥ α₂^g). In a larger, more irregular simplicial complex, the dynamics become more complex; the synchronization level decreases with increasing frustration, and the system exhibits chaotic behavior for large values of α₂. The largest Lyapunov exponent is used to confirm the transition into chaotic dynamics, revealing two distinct chaotic regimes. Furthermore, the exploration of weights in a simplicial complex consisting of two triangles shows that varying the weight of a face significantly alters the synchronization patterns, highlighting the importance of weights in modulating interactions.

The findings of this study significantly advance our understanding of synchronization in systems with higher-order interactions. The extension of the Sakaguchi-Kuramoto model to simplicial complexes provides a valuable tool for investigating complex systems where interactions are not limited to pairwise couplings. The hierarchical coupling between the harmonic, gradient, and curl subspaces, induced by frustration, reveals the intricate interplay between topology and dynamics. The sensitivity to edge orientation emphasizes the importance of considering directionality in complex networks. The observed transitions from synchronized to chaotic behavior highlight the richness of the dynamics, and the use of the largest Lyapunov exponent confirms the emergence of chaos. The incorporation of weights opens avenues to explore heterogeneous couplings and their influence on synchronization patterns. This work establishes a strong foundation for future research into higher-order synchronization phenomena, offering a framework for studying complex systems across diverse domains.

This paper presents a novel mathematical framework for studying coupled oscillators on simplicial complexes, incorporating weights and frustration terms. The model reveals complex dynamics influenced by topology and edge orientation, transitioning from synchronized to chaotic behavior. The Hodge decomposition provides a powerful tool for analyzing the system's behavior. Future work could explore the interplay between linear and nonlinear frustration, the role of weights in more detail, and the relationship between topological properties of simplicial complexes and the resulting dynamics. Investigating specific geometries and their potential to support particular synchronization patterns would also be a valuable area of future research.

The numerical simulations were conducted on specific examples of simplicial complexes. The generalizability of the observed dynamics to arbitrary complex structures needs further investigation. While the effect of weights was explored in a simple example, a systematic study encompassing various weight assignments and their impact on the dynamics is needed for a comprehensive understanding. The analysis focused on edge oscillators (k=1); exploration of higher-order simplices would provide further insights. The analysis of the transition to chaos was based on the Lyapunov exponents; a more in-depth analysis using other chaos indicators could provide further confirmation.