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Abstract

In this paper, we propose a D-dimensional topological Kuramoto model and investigate its synchronization on simplicial complexes. This model extends the higher-order Kuramoto model [Phys. Rev. Lett. 124, 218301 (2020)] to the D-dimensional sphere, where dynamics defined on simplices of different dimensions are governed by the D-dimensional Kuramoto model. By adopting an adaptive coupling, new phenomena of phase transitions are observed. Specifically, for nodal dynamics of odd dimensions, a double discontinuous synchronization transition is observed, whereas for the Kuramoto model defined on links, a single discontinuous synchronization transition occurs. Rigorous theoretical analysis reveals that the double discontinuous transition originates from the loss of stability of the incoherent state and a saddle-node bifurcation in the parameter space. Furthermore, for the D-dimensional Kuramoto model defined on links with D>2, synchronization is unattainable because of the inability to project dynamics onto adjacent dimensional simplices. Our findings provide insights into collective behaviors in high-dimensional spaces, such as collective defense mechanisms or social synchronization in insect swarms, mediated by higher-order signal transmission or environmental coupling.