  For gradient flows of energies, both spectral renormalization (SRN) and
energy landscape (EL) techniques have been used to establish slow motion of
orbits near low-energy manifold. We show that both methods are applicable to
flows induced by families of gradients and compare the scope and specificity of
the results. The SRN techniques capture the flow in a thinner neighborhood of
the manifold, affording a leading order representation of the slow flow via as
projection of the flow onto the tangent plane of the manifold. The SRN approach
requires a spectral gap in the linearization of the full gradient flow about
the points on the low-energy manifold. We provide conditions on the choice of
gradient under which the spectral gap is preserved, and show that up to
reparameterization the slow flow is invariant under these choices of gradients.
The EL methods estimate the magnitude of the slow flow, but cannot capture its
leading order form. However the EL only requires normal coercivity for the
second variation of the energy, and does not require spectral conditions on the
linearization of the full flow. It thus applies to a much larger class of
gradients of a given energy. We develop conditions under which the assumptions
of the SRN method imply the applicability of the EL method, and identify a
large family of gradients for which the EL methods apply. In particular we
apply both approaches to derive the interaction of multi-pulse solutions within
the 1+1D Functionalized Cahn-Hilliard (FCH) gradient flow, deriving gradient
invariance for a class of gradients arising from powers of a homogeneous
differential operator.
