Continuity and Stability of the FitzHugh Nagumo System with Respect to the Diffusion
Abstract
Spatial heterogeneity in diffusion is a fundamental feature of excitable media, yet its effect on FitzHugh Nagumo dynamics remains mathematically challenging. We establish stability of solutions with respect to perturbations of the diffusion coefficient, providing a rigorous account of how changes in the medium affect the evolution of the system. Specifically, we prove a continuous dependence result ensuring that small perturbations in diffusion produce small variations in the corresponding solutions. The proof adapts the method of contracting rectangles to variable diffusion, yielding a framework for controlling and comparing solutions associated with different diffusion profiles. This result provides a rigorous link between heterogeneous and homogeneous FitzHugh Nagumo systems and justifies constant-diffusion approximations when the diffusion coefficient is close to a constant. Our analysis advances the understanding of FitzHugh Nagumo dynamics in heterogeneous media by establishing robustness with respect to spatial variations in diffusion.
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BibTeX
@misc{rodriguez2026continuity,
title = {{Continuity and Stability of the FitzHugh Nagumo System with Respect to the Diffusion}},
author = {Alexander Rodriguez},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05770},
eprint = {2611.05770},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05770},
}
RIS
TY - UNPB
TI - Continuity and Stability of the FitzHugh Nagumo System with Respect to the Diffusion
AU - Alexander Rodriguez
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05770
UR - https://arxiv.org/abs/2611.05770
AB - Spatial heterogeneity in diffusion is a fundamental feature of excitable media, yet its effect on FitzHugh Nagumo dynamics remains mathematically challenging. We establish stability of solutions with respect to perturbations of the diffusion coefficient, providing a rigorous account of how changes in the medium affect the evolution of the system. Specifically, we prove a continuous dependence result ensuring that small perturbations in diffusion produce small variations in the corresponding solutions. The proof adapts the method of contracting rectangles to variable diffusion, yielding a framework for controlling and comparing solutions associated with different diffusion profiles. This result provides a rigorous link between heterogeneous and homogeneous FitzHugh Nagumo systems and justifies constant-diffusion approximations when the diffusion coefficient is close to a constant. Our analysis advances the understanding of FitzHugh Nagumo dynamics in heterogeneous media by establishing robustness with respect to spatial variations in diffusion.
ER -