Abstract
We develop general methods for rigorously computing continuous branches of bifurcation points of equilibria, specifically focusing on fold points and on pitchfork bifurcations which are forced through \begin{document}${\mathbb{Z}}_2$\end{document} symmetries in the equation. We apply these methods to secondary bifurcation points of the one-dimensional diblock copolymer model.
Highlights
Understanding the equilibrium structure of nonlinear partial differential equations lies at the heart of many important applied problems
One alternative that has been developed over the last decades is the use of computer-assisted proofs in the study of equilibrium problems of nonlinear partial differential equations
We develop a generally applicable method for solving problems with saddle-node bifurcations and Z2 symmetry-breaking pitchfork bifurcations
Summary
Understanding the equilibrium structure of nonlinear partial differential equations lies at the heart of many important applied problems. While this could lead to a situation where no bifurcation occurs, for example if the solution curve is monotone with respect to α, by assuming an additional generic condition one can guarantee a true saddle-node bifurcation point.
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