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In this paper, we combine the method of multiple scales and the method ofmatched asymptotic expansions to construct uniformly-valid asymptotic solutionsto autonomous and non-autonomous difference equations in the neighbourhood of aperiod-doubling bifurcation. In each case, we begin by constructing multiplescales approximations in which the slow time scale is treated as a continuumvariable, leading to difference-differential equations. The resultantapproximations fail to be asymptotic at late time, due to behaviour on the slowtime scale, it is necessary to eliminate the effects of the fast time scale inorder to find the late time rescaling, but there are then no difficulties withapplying the method of matched asymptotic expansions. The methods that wedevelop lead to a general strategy for obtaining asymptotic solutions tosingularly-perturbed difference equations, and we discuss clear indicators ofwhen multiple scales, matched asymptotic expansions, or a combined approachmight be appropriate.

Multiple scales and matched asymptotic expansions for the discrete logistic equation