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*Title : A general theory of phase reduction for systems with higher-order interactions*

Synchronization is a ubiquitous emergent phenomenon in which an ensemble of elementary units behaves in unison due to their interactions [1]. Given the pervasiveness of synchronization, understanding how it is achieved is a fundamental question. In particular, the nature of the interactions among oscillators has strong consequences on the transition to synchronization. To tackle this issue, it is convenient to consider phase models in which each oscillator is described solely in terms of a phase variable. According to phase reduction theory, the phase model captures the dynamics completely when the coupling among the oscillators is sufficiently weak [2]. If one considers only pairwise interactions, the synchronization transition is described by the Kuramoto-type model. Despite the versatility of such an approach, the classical theory of synchronization is solely based on pairwise interactions, while, in many natural systems, the interactions are intrinsically higher-order (many-body) rather than pairwise [3]. In fact, many examples show that a pairwise description is sufficient not to match the theory with observations and, additionally, higher-order interactions appear naturally when phase reduction is performed up to higher orders [4]. It was also shown that extensions of the Kuramoto model including higher-order interactions exhibit an explosive transition and other interesting dynamics [5].

This seminar will be divided in two parts. In the first part, I will introduce the phase reduction theory and highlight the universality of phase models and the generality of the approach, especially when compared to the Master Stability Function [6]. Then, after discussing the basics of higher-order interactions, I will present a recent work where we analyzed the collective dynamics of the simplest minimal extension of the Kuramoto-type phase model for identical globally coupled oscillators subject to two- and three-body interactions and showed how the many-body interactions greatly enriches the behaviors of the system [7]. Lastly, I will show some preliminary results for some general cases, one where the coupling is not constrained to the all-to-all case [8] and the other in which the phase reduction cannot be performed analytically [9].

This is a joint work with Hiroya Nakao (Tokyo Institute of Technology, Japan), Iván León (Universidad de Cantabria, Spain) and Shigefumi Hata (Kagoshima University, Japan)

References

[1] Kuramoto Y., Chemical Oscillations, Waves, and Turbulence. Springer-Verlag, 1984.

[2] Nakao H., Phase reduction approach to synchronisation of nonlinear oscillators. Cont. Phys., 57(2): 188-214, 2016.

[3] Battiston F. et al., Networks beyond pairwise interactions: Structure and dynamics. Phys. Rep., 84: 1-92, 2020.

[4] León I. and Pazó D., Phase reduction beyond the first order: The case of the mean-field complex Ginzburg-Landau equation. Phys. Rev. E, 100(1): 012211, 2019.

[5] Skardal P.S. and Arenas A., Abrupt Desynchronization and Extensive Multistability in Globally Coupled Oscillator Simplexes. Phys. Rev. Lett., 122(84): 248301, 2019.

[6] Fujisaka, H., and Yamada, T., Stability theory of synchronized motion in coupled-oscillator systems. Prog. Theor. Phys., 69(1), 32-47, 1983.

[7] León I., Muolo R., Hata S. and Nakao H., Higher-order interactions induce anomalous transitions to synchrony. To appear in Chaos, Dec 2023.

[8] Muolo R., León I., Hata S. and Nakao H., Phase Reduction Analysis of Collective Dynamics in Systems of Coupled Oscillators with Higher-Order Interactions. In preparation.

[9] Muolo R. and Nakao H., A general framework to study spiking neurons coupled through higher-order interactions. In preparation.

The seminar will take place in **Room S08** at the Faculty of Sciences.