  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.
