Phase locking and multistability in the topological Kuramoto model on cell complexes
Nature Communications, vol. 17
Abstract
Higher-order interactions fundamentally shape collective dynamics in oscillator networks. The topological Kuramoto model captures these effects by extending synchronization models to include interactions between cells of arbitrary dimension within simplicial and cell complexes. We introduce the topological nonlinear Kirchhoff conditions to characterize all phase-locked states of the topological Kuramoto model. These states are organized by winding numbers associated with generalized independent cycles, which quantify how phases wind around these cycles. Using rings, Platonic solids, and regular simplices as illustrative examples, we uncover a universal rule: boundaries must have at least five elements for multistability to arise. We further find that independent winding numbers associated with lower- and higher-dimensional boundaries generate cascades of multistability across dimensions. These results show how the topology and boundary structure of cell complexes influence phase locking and multistability, and provide a general framework for collective dynamics on cell complexes.
Authors 4
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Iva Bačić corresponding
Forschungszentrum Jülich · University of Belgrade · Institute of Physics Belgrade
Affiliation as printed
Center of Excellence for Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Belgrade, Serbia. i.bacic@fz-juelich.de
Institute of Climate and Energy Systems: Energy Systems Engineering (ICE-1), Forschungszentrum Jülich, Jülich, Germany. i.bacic@fz-juelich.de
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Michael T. Schaub Aachen
Affiliation as printed
RWTH Aachen University, Aachen, Germany
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Potsdam Institute for Climate Impact Research
Affiliation as printed
Potsdam Institute for Climate Impact Research (PIK), Potsdam, Germany
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Dirk Witthaut corresponding
Forschungszentrum Jülich · University of Cologne
Affiliation as printed
Institute for Theoretical Physics, University of Cologne, Köln, Germany. d.witthaut@fz-juelich.de
Institute of Climate and Energy Systems: Energy Systems Engineering (ICE-1), Forschungszentrum Jülich, Jülich, Germany. d.witthaut@fz-juelich.de
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