Diffusive synchronization of phase waves in the FitzHugh-Nagumo system
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| Format: | Preprint |
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2026
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| author | Avery, Montie Carter, Paul de Rijk, Björn Scheel, Arnd |
| author_facet | Avery, Montie Carter, Paul de Rijk, Björn Scheel, Arnd |
| contents | We analyze synchronization of relaxation oscillations in multiple-timescale reaction-diffusion systems. Interpreting synchronization as convergence to frequency-synchronized wave-train solutions, we resolve for the first time the case of phase waves. These waves are nearly phase-synchronized relaxation oscillations, featuring quasistationary plateaus of length $\varepsilon^{-1}$ separated by fast transition layers, where $\varepsilon\ll1$ is the timescale separation parameter. Tracking the decay of modulations via a Bloch-wave eigenfunction analysis, we find a remarkably weak interaction strength of order $\varepsilon^{8/3}$. This weak layer interaction and many of the technical difficulties arise from repeated scattering of eigenfunctions through fold points at the ends of the quasistationary plateaus. We capture this by combining a novel geometric desingularization approach with Lin's method, exponential trichotomies, and the Riccati transform. While our spectral stability analysis yields diffusive synchronization of all phase waves in the FitzHugh-Nagumo system, it also identifies potential finite-wavelength instabilities, which we realize in a system variant. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_05377 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Diffusive synchronization of phase waves in the FitzHugh-Nagumo system Avery, Montie Carter, Paul de Rijk, Björn Scheel, Arnd Analysis of PDEs Dynamical Systems Pattern Formation and Solitons 35B10, 35B25, 35B35, 35K57, 35P15 We analyze synchronization of relaxation oscillations in multiple-timescale reaction-diffusion systems. Interpreting synchronization as convergence to frequency-synchronized wave-train solutions, we resolve for the first time the case of phase waves. These waves are nearly phase-synchronized relaxation oscillations, featuring quasistationary plateaus of length $\varepsilon^{-1}$ separated by fast transition layers, where $\varepsilon\ll1$ is the timescale separation parameter. Tracking the decay of modulations via a Bloch-wave eigenfunction analysis, we find a remarkably weak interaction strength of order $\varepsilon^{8/3}$. This weak layer interaction and many of the technical difficulties arise from repeated scattering of eigenfunctions through fold points at the ends of the quasistationary plateaus. We capture this by combining a novel geometric desingularization approach with Lin's method, exponential trichotomies, and the Riccati transform. While our spectral stability analysis yields diffusive synchronization of all phase waves in the FitzHugh-Nagumo system, it also identifies potential finite-wavelength instabilities, which we realize in a system variant. |
| title | Diffusive synchronization of phase waves in the FitzHugh-Nagumo system |
| topic | Analysis of PDEs Dynamical Systems Pattern Formation and Solitons 35B10, 35B25, 35B35, 35K57, 35P15 |
| url | https://arxiv.org/abs/2601.05377 |