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We consider the problem of ignition of propagating waves in one-dimensionalbistable 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 anapproximation 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 withnon-self-adjoint linearized operators, including systems with moving criticalfronts and pulses. We also explore an extension of this method from a linear toa quadratic approximation of the (center-)stable manifold, resulting in somecases in a significant increase in accuracy. The applicability of the approachis demonstrated on five test problems ranging from archetypal examples such asthe Zeldovich--Frank-Kamenetsky equation to near realistic examples such as theBeeler-Reuter model of cardiac excitation. While the method is analytical innature, it is recognised that essential ingredients of the theory can becalculated explicitly only in exceptional cases, so we also describe methodssuitable for calculating these ingredients numerically.

Semi-analytical approach to criteria for ignition of excitation waves