Capturing the cascade: a transseries approach to delayed bifurcations

Inês Aniceto*, Daniel Hasenbichler, Christopher J. Howls, Christopher J. Lustri

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    3 Citations (Scopus)
    37 Downloads (Pure)

    Abstract

    Transseries expansions build upon ordinary power series methods by including additional basis elements such as exponentials and logarithms. Alternative summation methods can then be used to 'resum' series to obtain more efficient approximations, and have been successfully widely applied in the study of continuous linear and nonlinear, single and multidimensional problems. In particular, a method known as transasymptotic resummation can be used to describe continuous behaviour occurring on multiple scales without the need for asymptotic matching. Here we apply transasymptotic resummation to discrete systems and show that it may be used to naturally and efficiently describe discrete delayed bifurcations, or 'canards', in singularly-perturbed variants of the logistic map which contain delayed period-doubling bifurcations. We use transasymptotic resummation to approximate the solutions, and describe the behaviour of the solution across the bifurcations. This approach has two significant advantages: it may be applied in systematic fashion even across multiple bifurcations, and the exponential multipliers encode information about the bifurcations that are used to explain effects seen in the solution behaviour.

    Original languageEnglish
    Pages (from-to)8248-8282
    Number of pages35
    JournalNonlinearity
    Volume34
    Issue number12
    DOIs
    Publication statusPublished - Dec 2021

    Bibliographical note

    Copyright the Publisher 2021. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

    Keywords

    • logistic equation
    • transseries
    • delayed bifurcations
    • series summation
    • difference equation

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