Multiple scales and matched asymptotic expansions for the discrete logistic equation

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• 作者：Cameron L. Hall ; Christopher J. Lustri
• 刊名：Nonlinear Dynamics
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：85
• 期：2
• 页码：1345-1362
• 全文大小：1,178 KB
• 刊物类别：Engineering
• 刊物主题：Vibration, Dynamical Systems and Control
  Mechanics
  Mechanical Engineering
  Automotive and Aerospace Engineering and Traffic
  
• 出版者：Springer Netherlands
• ISSN：1573-269X
• 卷排序：85

文摘

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.KeywordsDifference equationsAsymptotic analysisDiscrete-to-continuum asymptoticsLogistic difference equation