Probing a subcritical instability with an amplitude expansion: how far can one get?

We evaluate a method to describe subcritical transitions in systems with a finitewavelength instability (pattern forming systems) by means of a direct expansion in the amplitude of the linearly least stable mode. We apply the method to two model equations, a subcritical generalization of the Swift-Hohenberg equation and an extension of the Kuramoto-Sivashinsky equation. We assess the reliability and robustness of such an expansion, with a particular focus on the use of these methods for determining the existence and approximate properties of finite-amplitude stationary solutions. Such methods obviously are to be used with caution: the expansions are often only asymptotic approximations, and if they converge their radius of convergence may be small. Nevertheless, expansions to higher order in the amplitude can be a useful tool to obtain qualitatively reliable results.