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Lee S, Braun L, Bönisch F, Schröder M, Thümler M, Timme M. Complexified synchrony. CHAOS (WOODBURY, N.Y.) 2024; 34:053141. [PMID: 38814675 DOI: 10.1063/5.0205897] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/28/2024] [Accepted: 05/06/2024] [Indexed: 05/31/2024]
Abstract
The Kuramoto model and its generalizations have been broadly employed to characterize and mechanistically understand various collective dynamical phenomena, especially the emergence of synchrony among coupled oscillators. Despite almost five decades of research, many questions remain open, in particular, for finite-size systems. Here, we generalize recent work [Thümler et al., Phys. Rev. Lett. 130, 187201 (2023)] on the finite-size Kuramoto model with its state variables analytically continued to the complex domain and also complexify its system parameters. Intriguingly, systems of two units with purely imaginary coupling do not actively synchronize even for arbitrarily large magnitudes of the coupling strengths, |K|→∞, but exhibit conservative dynamics with asynchronous rotations or librations for all |K|. For generic complex coupling, both traditional phase-locked states and asynchronous states generalize to complex locked states, fixed points off the real subspace that exist even for arbitrarily weak coupling. We analyze a new collective mode of rotations exhibiting finite, yet arbitrarily large rotation numbers. Numerical simulations for large networks indicate a novel form of discontinuous phase transition. We close by pointing to a range of exciting questions for future research.
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Affiliation(s)
- Seungjae Lee
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Lucas Braun
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
- Schülerforschungszentrum Südwürttemberg (SFZ), 88348 Bad Saulgau, Germany
- Gymnasium Wilhelmsdorf, Pfrunger Straße 4/2, 88271 Wilhelmsdorf, Germany
| | - Frieder Bönisch
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Malte Schröder
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Moritz Thümler
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Marc Timme
- Chair for Network Dynamics, Center for Advancing Electronics Dresden (CFAED) and Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
- Cluster of Excellence Physics of Life, TU Dresden, 01062 Dresden, Germany
- Lakeside Labs, 9020 Klagenfurt, Austria
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de Aguiar MAM. Generalized frustration in the multidimensional Kuramoto model. Phys Rev E 2023; 107:044205. [PMID: 37198798 DOI: 10.1103/physreve.107.044205] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Accepted: 03/24/2023] [Indexed: 05/19/2023]
Abstract
The Kuramoto model describes how coupled oscillators synchronize their phases as the intensity of the coupling increases past a threshold. The model was recently extended by reinterpreting the oscillators as particles moving on the surface of unit spheres in a D-dimensional space. Each particle is then represented by a D-dimensional unit vector; for D=2 the particles move on the unit circle and the vectors can be described by a single phase, recovering the original Kuramoto model. This multidimensional description can be further extended by promoting the coupling constant between the particles to a matrix K that acts on the unit vectors. As the coupling matrix changes the direction of the vectors, it can be interpreted as a generalized frustration that tends to hinder synchronization. In a recent paper we studied in detail the role of the coupling matrix for D=2. Here we extend this analysis to arbitrary dimensions. We show that, for identical particles, when the natural frequencies are set to zero, the system converges either to a stationary synchronized state, given by one of the real eigenvectors of K, or to an effective two-dimensional rotation, defined by one of the complex eigenvectors of K. The stability of these states depends on the set eigenvalues and eigenvectors of the coupling matrix, which controls the asymptotic behavior of the system, and therefore, can be used to manipulate these states. When the natural frequencies are not zero, synchronization depends on whether D is even or odd. In even dimensions the transition to synchronization is continuous and rotating states are replaced by active states, where the module of the order parameter oscillates while it rotates. If D is odd the phase transition is discontinuous and active states can be suppressed for some distributions of natural frequencies.
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Affiliation(s)
- Marcus A M de Aguiar
- Instituto de Física "Gleb Wataghin", Universidade Estadual de Campinas, Unicamp 13083-970, Campinas, São Paolo, Brazil
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