  We consider the problem of ignition of propagating waves in one-dimensional
bistable or excitable systems by an instantaneous spatially extended stimulus.
Earlier we proposed a method (Idris and Biktashev, PRL, vol 101, 2008, 244101)
for analytical description of the threshold conditions based on an
approximation of the (center-)stable manifold of a certain critical solution.
Here we generalize this method to address a wider class of excitable systems,
such as multicomponent reaction-diffusion systems and systems with
non-self-adjoint linearized operators, including systems with moving critical
fronts and pulses. We also explore an extension of this method from a linear to
a quadratic approximation of the (center-)stable manifold, resulting in some
cases in a significant increase in accuracy. The applicability of the approach
is demonstrated on five test problems ranging from archetypal examples such as
the Zeldovich--Frank-Kamenetsky equation to near realistic examples such as the
Beeler-Reuter model of cardiac excitation. While the method is analytical in
nature, it is recognised that essential ingredients of the theory can be
calculated explicitly only in exceptional cases, so we also describe methods
suitable for calculating these ingredients numerically.
