Symmetry induced pitchfork and transcritical bifurcations in nonlinear diffusion problems

84. Maxime Breden, Evelyn Sander, Thomas Wanner:
    Symmetry induced pitchfork and transcritical bifurcations in nonlinear diffusion problems
    Preprint, submitted for publication, 46 pages, 2026.

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 is 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.

Bibtex

@article{breden:sander:wanner:p26a,
   author = {Maxime Breden and Evelyn Sander and Thomas Wanner},
   title = {Symmetry induced pitchfork and transcritical
            bifurcations in nonlinear diffusion problems},
   journal = {Submitted for publication},
   year = 2026
   }