Multiple scales and matched asymptotic expansions for the discrete logistic equation

In this paper, we combine the method of multiple scales and the method of matched asymptotic expansions to construct uniformly valid asymptotic solutions to autonomous and non-autonomous forms of the discrete logistic equation in the neighbourhood of a period-doubling bifurcation. In each case, we begin by constructing a multiple scales approximation in which the fast time scale is treated as discrete, but the slow time scale is treated as continuous. The resulting multiple scales solutions are initially accurate, but fail to be asymptotic at late times due to changes in dominant balance that occur on the slow time scale. We address these problems by determining the variable rescalings associated with the late-time distinguished limit and applying the method of matched asymptotic expansions. This process leads to novel uniformly valid asymptotic solutions that could not have been obtained using the method of multiple scales or the method of matched asymptotic expansions alone. While we concentrate on the discrete logistic equation throughout, the methods that we develop lead to general strategies for obtaining asymptotic solutions to singularly perturbed difference equations, and we discuss clear indicators of when multiple scales, matched asymptotic expansions, or a combined approach might be appropriate.