  The functionalized Cahn-Hilliard (FCH) equation supports planar and circular
bilayer interfaces as equilibria which may lose their stability through the
pearling bifurcation: a periodic, high-frequency, in-plane modulation of the
bilayer thickness. In two spatial dimensions we employ spatial dynamics and a
center manifold reduction to reduce the FCH equation to an 8th order ODE
system. A normal form analysis and a fixed-point-theorem argument show that the
reduced system admits a degenerate 1:1 resonant normal form, from which we
deduce that the onset of the pearling bifurcation coincides with the creation
of a two-parameter family of pearled equilibria which are periodic in the
in-plane direction and exponentially localized in the transverse direction.
