Validation of Symmetry Induced Bifurcation Points in Nonlinear Diffusion Problems
Abstract
Nonlinear parabolic partial differential equations frequently occur in the modeling of applied phenomena. They often exhibit complicated dynamics which is organized by their associated equilibrium solutions. As a function of the involved model parameters, the equilibria can be detected using a bifurcation analysis, which in turn is centered around a deeper understanding of bifurcation points. In many applied systems establishing these organizing centers rigorously is difficult to achieve using traditional mathematical techniques. In this paper, we propose a theoretical framework for computer-assisted proofs of symmetry-induced transcritical and pitchfork bifurcations in nonlinear problems. This s accomplished by reformulating the existence question via a zero finding problem for an associated extended nonlinear system. The approach is general and allows for a wide variety of underlying symmetries. We demonstrate how these results can be applied in the setting of parabolic systems and higher-order equations, on both one- and two-dimensional domains.
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BibTeX
@misc{breden2026validation,
title = {{Validation of Symmetry Induced Bifurcation Points in Nonlinear Diffusion Problems}},
author = {Maxime Breden and Evelyn Sander and Thomas Wanner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38629},
eprint = {2609.38629},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38629},
}
RIS
TY - UNPB
TI - Validation of Symmetry Induced Bifurcation Points in Nonlinear Diffusion Problems
AU - Maxime Breden
AU - Evelyn Sander
AU - Thomas Wanner
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38629
UR - https://arxiv.org/abs/2609.38629
AB - Nonlinear parabolic partial differential equations frequently occur in the modeling of applied phenomena. They often exhibit complicated dynamics which is organized by their associated equilibrium solutions. As a function of the involved model parameters, the equilibria can be detected using a bifurcation analysis, which in turn is centered around a deeper understanding of bifurcation points. In many applied systems establishing these organizing centers rigorously is difficult to achieve using traditional mathematical techniques. In this paper, we propose a theoretical framework for computer-assisted proofs of symmetry-induced transcritical and pitchfork bifurcations in nonlinear problems. This s accomplished by reformulating the existence question via a zero finding problem for an associated extended nonlinear system. The approach is general and allows for a wide variety of underlying symmetries. We demonstrate how these results can be applied in the setting of parabolic systems and higher-order equations, on both one- and two-dimensional domains.
ER -