Geometric perspective of linear stability in finite networks of nonlinear oscillators
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arXiv
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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910906939604992 |
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| author | Sinha, Yashee Jain, Priya B. Mihara, Antonio Medrano-T, Rene O. Mináč, Ján Muller, Lyle E. Budzinski, Roberto C. |
| author_facet | Sinha, Yashee Jain, Priya B. Mihara, Antonio Medrano-T, Rene O. Mináč, Ján Muller, Lyle E. Budzinski, Roberto C. |
| contents | We use a complex-valued transformation of the Kuramoto model to develop an operator-description of the linear stability in finite networks of nonlinear oscillators. This mathematical approach offers analytical predictions for the linear stability of $q$-states, which include phase synchronization ($q = 0$) and waves with different spatial frequencies ($|q| > 0$). This approach seamlessly incorporates the presence of time delays (represented by phase-lags in the coupling). With this, we are able to analytically determine the specific combination of connectivity and time delays (phase-lags) that leads to any given $q$-state to be linearly stable. This approach offers a geometric perspective of linear stability in finite networks in terms of the connectivity and delays (phase-lag), and it opens a path to designing and controlling the spatiotemporal dynamics of individual oscillator networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06377 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Geometric perspective of linear stability in finite networks of nonlinear oscillators Sinha, Yashee Jain, Priya B. Mihara, Antonio Medrano-T, Rene O. Mináč, Ján Muller, Lyle E. Budzinski, Roberto C. Dynamical Systems We use a complex-valued transformation of the Kuramoto model to develop an operator-description of the linear stability in finite networks of nonlinear oscillators. This mathematical approach offers analytical predictions for the linear stability of $q$-states, which include phase synchronization ($q = 0$) and waves with different spatial frequencies ($|q| > 0$). This approach seamlessly incorporates the presence of time delays (represented by phase-lags in the coupling). With this, we are able to analytically determine the specific combination of connectivity and time delays (phase-lags) that leads to any given $q$-state to be linearly stable. This approach offers a geometric perspective of linear stability in finite networks in terms of the connectivity and delays (phase-lag), and it opens a path to designing and controlling the spatiotemporal dynamics of individual oscillator networks. |
| title | Geometric perspective of linear stability in finite networks of nonlinear oscillators |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2504.06377 |