Equivalence of Synchronization States in the Hybrid Kuramoto Flow
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909946653704192 |
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| author | Hsiao, Ting-Yang Lo, Yun-Feng Zhu, Chengbin |
| author_facet | Hsiao, Ting-Yang Lo, Yun-Feng Zhu, Chengbin |
| contents | We establish a unified synchronization framework for the all-to-all hybrid Kuramoto model that couples first- and second-order oscillators within a single dynamical system. Although the Kuramoto model has become one of the most widely used paradigms for describing synchronization phenomena-appearing in more than 100,000 scientific studies-the fundamental relationships among distinct synchronization states remain unresolved. In this work, we rigorously prove that full phase-locking, phase-locking, frequency synchronization, and order-parameter synchronization are equivalent for arbitrary hybrid ensembles. The proof combines dissipative energy methods, LaSalle-type compactness arguments, the Poincar{é}-Bendixson theorem, and Thieme's asymptotically autonomous theory to demonstrate that synchronization equivalence is topological, determined solely by the finite equilibrium structure of the all-to-all network. This result provides a complete mathematical characterization of synchronization in finite oscillator systems and clarifies its geometric invariance across first-, second-, and hybrid-order models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02986 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equivalence of Synchronization States in the Hybrid Kuramoto Flow Hsiao, Ting-Yang Lo, Yun-Feng Zhu, Chengbin Dynamical Systems Primary 34D45, Secondary 34D06, 34C25, 37C10 We establish a unified synchronization framework for the all-to-all hybrid Kuramoto model that couples first- and second-order oscillators within a single dynamical system. Although the Kuramoto model has become one of the most widely used paradigms for describing synchronization phenomena-appearing in more than 100,000 scientific studies-the fundamental relationships among distinct synchronization states remain unresolved. In this work, we rigorously prove that full phase-locking, phase-locking, frequency synchronization, and order-parameter synchronization are equivalent for arbitrary hybrid ensembles. The proof combines dissipative energy methods, LaSalle-type compactness arguments, the Poincar{é}-Bendixson theorem, and Thieme's asymptotically autonomous theory to demonstrate that synchronization equivalence is topological, determined solely by the finite equilibrium structure of the all-to-all network. This result provides a complete mathematical characterization of synchronization in finite oscillator systems and clarifies its geometric invariance across first-, second-, and hybrid-order models. |
| title | Equivalence of Synchronization States in the Hybrid Kuramoto Flow |
| topic | Dynamical Systems Primary 34D45, Secondary 34D06, 34C25, 37C10 |
| url | https://arxiv.org/abs/2512.02986 |