Cyclic symmetry induced pitchfork bifurcations in the diblock copolymer model

TL;DR

This paper investigates complex bifurcation structures in the diblock copolymer model, revealing cyclic symmetry-induced pitchfork bifurcations and demonstrating how computer-assisted proofs can validate these bifurcations.

Contribution

It introduces a novel approach linking cyclic group actions to pitchfork bifurcations and extends proof techniques to validate these bifurcations in complex systems.

Findings

Cyclic groups of even order induce pitchfork bifurcations.

Bifurcation points are characterized as nondegenerate solutions of extended systems.

Computer-assisted proofs can validate complex bifurcations in the model.

Abstract

The Ohta-Kawasaki model for diblock copolymers exhibits a rich equilibrium bifurcation structure. Even on one-dimensional base domains the bifurcation set is characterized by high levels of multi-stability and numerous secondary bifurcation points. Many of these bifurcations are of pitchfork type. In previous work, the authors showed that if pitchfork bifurcations are induced by a simple $\mathbb{Z}_2$ symmetry-breaking, then computer-assisted proof techniques can be used to rigorously validate them using extended systems. However, many diblock copolymer pitchfork bifurcations cannot be treated in this way. In the present paper, we show that in these more involved cases, a cyclic group action is responsible for their existence, based on cyclic groups of even order. We present theoretical results establishing such bifurcation points and show that they can be characterized as nondegenerate solutions of a suitable extended nonlinear system. Using the latter characterization, we also demonstrate that computer-assisted proof techniques can be used to validate such bifurcations. While the methods proposed in this paper are only applied to the diblock copolymer model, we expect that they will also apply to other parabolic partial differential equations.