Multiple scales and matched asymptotic expansions for the discrete logistic equation

摘要

In this paper, we combine the method ofmultiple scales and the method of matched asymptoticexpansions to construct uniformly-valid asymptotic solutionsto autonomous and non-autonomous forms ofthe discrete logistic equation in the neighbourhood of aperiod-doubling bifurcation. In each case, we begin byconstructing a multiple scales approximation in whichthe fast time scale is treated as discrete but the slowtime scale is treated as continuous. The resulting multiplescales solutions are initially accurate, but fail tobe asymptotic at late times due to changes in dominantbalance that occur on the slow time scale. We addressthese problems by determining the variable rescalingsassociated with the late-time distinguished limitand applying the method of matched asymptotic expansions.This process leads to novel uniformly-validasymptotic solutions that could not have been obtainedusing the method of multiple scales or the method ofmatched asymptotic expansions alone. While we concentrateon the discrete logistic equation throughout,the methods that we develop lead to general strategiesfor obtaining asymptotic solutions to singularlyperturbeddifference equations, and we discuss clear indicatorsof when multiple scales, matched asymptoticexpansions, or a combined approach might be appropriate