We are excited to host Ricardo Muolo who will present in the BeyondTheEdge Seminar series with a talk titled A phase reduction theory of coupled oscillators with nonpairwise interactions
Synchronization is a ubiquitous phenomenon observed in many natural and engineered systems. A standard framework for the study of synchronization is the Kuramoto model [1], which is obtained by phase reduction of weakly coupled nonlinear oscillators [2]. In its classical formulation, the model assumes pairwise interactions between oscillators.
Recent studies have shown that many systems are more naturally described by interactions involving three or more units simultaneously. This has motivated the extension of network theory to nonpairwise (higher-order) structures such as hypergraphs and simplicial complexes [3], as well as the development of nonpairwise extensions of the Kuramoto model. These models exhibit synchronization phenomena that are absent in the pairwise case, including multistability and cluster states [4,5], together with nontrivial effects on the basins of attraction of synchronous states [6,7].
A natural question is then how nonpairwise interactions arise in coupled oscillator systems and which forms of nonpairwise coupling are physically meaningful. In this talk, I will discuss how nonpairwise phase models can be systematically derived from coupled nonlinear oscillators using phase reduction. I will first review how nonpairwise interactions emerge through higher-order phase reduction [8] and compare the resulting interaction terms with the physical interaction terms already present in the original oscillator equations [9]. I will then present a general theory that derives higher-order Kuramoto models, first from normal forms [10] and then from arbitrary nonpairwise coupling structures [11]. Finally, I will discuss how symmetry arguments constrain the interaction terms appearing in the reduced dynamics, providing a systematic framework for identifying physically meaningful nonpairwise Kuramoto models [12].
Bio: Riccardo studied Physics (BSc) and Applied Mathematics (MSc) in Florence (Italy), before moving to Namur (Belgium) for his Master thesis and PhD in the group of Teo Carletti. He then moved to Tokyo Tech (now become institute of Science Tokyo) as a Postdoc in the group of Hiroya Nakao. Since last year, he joined the RIKEN Center for Interdisciplinary Mathematical and Theoretical Sciences (iTHEMS), where he's a Postdoc. His main interests are the study and control of dynamics on networks and higher-order structures, with particular focus on reduction methods for oscillatory systems.
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https://us06web.zoom.us/j/81698732992?pwd=qYsM8hskt9LZ74fuTpBkcemYzjrA4F.1
Meeting ID: 816 9873 2992
Passcode: 133958
References
[1] Kuramoto, Y. Chemical Oscillations, Waves, and Turbulence. Springer, 1984.
[2] Nakao, H. Phase reduction approach to synchronisation of nonlinear oscillators. Contemporary Physics 57(2), 188–214 (2016).
[3] Battiston, F., Cencetti, G., Iacopini, I., Latora, V., Lucas, M., Patania, A., Young, J.-G., & Petri, G. Networks beyond pairwise interactions: Structure and dynamics. Physics Reports 874, 1–92 (2020).
[4] Tanaka, T., & Aoyagi, T. Multistable attractors in a network of phase oscillators with three-body interactions. Physical Review Letters 106, 224101 (2011).
[5] Skardal, P. S., & Arenas, A. Higher-order interactions can better optimize network synchronization. Communications Physics 3, 218 (2020).
[6] Zhang, Y., Skardal, P. S., Battiston, F., Petri, G., and Lucas, M. Deeper but smaller: Higher-order interactions increase linear stability but shrink basins. Sci. Adv. 10, eado8049 (2024).
[7] Muolo, R., Nakao, H., & Coraggio, M. When Higher-Order Interactions Enhance Synchronization: The Case of the Kuramoto Model. arXiv:2508.10992 (2025).
[8] León, I., & Pazó, D. Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation, Phys. Rev. E 100, 012211 (2019).
[9] Muolo, R., Nakao, H., & Bick, C. Physical and emergent nonpairwise interactions in phase-reduced oscillator networks. In preparation.
[10] León, I., Muolo, R., Hata, S., & Nakao, H. Higher-order interactions induce anomalous transitions to synchrony. Chaos 34, 013105 (2024).
[11] León, I., Muolo, R., Hata, S., & Nakao, H. Theory of phase reduction from hypergraphs to simplicial complexes: A general route to higher-order Kuramoto models. Physica D 482, 134858 (2025).
[12] León, I., Muolo, R., Zhang, Y., & Lucas, M. Symmetry-based selection rules for higher-order interactions in coupled oscillators. arXiv:2606.04904 (2026).