  We consider two parameter families of differential equations on a Banach
space X, where the parameters c and $\epsilon$ are such that: (1) when
$\epsilon=0$, the differential equations are symmetric under the action of the
group of one-dimensional translations SE(1) acting on X, whereas when
$\epsilon\neq 0$, this translation symmetry is broken, (2) when $\epsilon=0$,
the symmetric differential equations admit a smooth family of relative
equilibria (travelling waves) parametrized by the drift speed c, with $c=0$
corresponding to steady-states. Under certain hypotheses on the differential
equations and on the Banach space X, we use the center manifold theorem of
Sandstede, Scheel and Wulff to study the effects of the symmetry-breaking
perturbation on the above family of relative equilibria. In particular, we show
that the phenomenon commonly referred to as propagation failure, or wave
blocking occurs in a cone in the $(c,\epsilon)$ parameter space which emanates
from the point $(c,\epsilon)=(0,0)$. We also discuss how our methods can be
adapted to perturbations of parameter-independent differential equations (such
as the Fisher-KPP) which admit families of relative equilibria parametrized by
drift speed.
