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Capturing the cascade: a transseries approach to delayed bifurcations

Capturing the cascade: a transseries approach to delayed bifurcations
Capturing the cascade: a transseries approach to delayed bifurcations
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.
delayed bifurcations, difference equation, logistic equation, series summation, transseries
0951-7715
8248
Aniceto, Ines
0061ca0c-1ad8-4510-9b12-008e5c27a7ea
Hasenbichler, Daniel
33471eb8-7e5c-4dd7-9062-6fed61ac9abe
Lustri, Christopher
fca62c45-bfaf-4fa4-a1d5-ddd55411eaef
Howls, Christopher
66d3f0f0-376c-4f7a-a206-093935e6c560
Aniceto, Ines
0061ca0c-1ad8-4510-9b12-008e5c27a7ea
Hasenbichler, Daniel
33471eb8-7e5c-4dd7-9062-6fed61ac9abe
Lustri, Christopher
fca62c45-bfaf-4fa4-a1d5-ddd55411eaef
Howls, Christopher
66d3f0f0-376c-4f7a-a206-093935e6c560

Aniceto, Ines, Hasenbichler, Daniel, Lustri, Christopher and Howls, Christopher (2021) Capturing the cascade: a transseries approach to delayed bifurcations. Nonlinearity, 34 (12), 8248. (doi:10.1088/1361-6544/ac2e44).

Record type: Article

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.

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2012.09779 - Accepted Manuscript
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Available under License Creative Commons Attribution.
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More information

Accepted/In Press date: 17 December 2020
e-pub ahead of print date: 17 December 2020
Published date: December 2021
Additional Information: Publisher Copyright: © 2021 IOP Publishing Ltd & London Mathematical Society.
Keywords: delayed bifurcations, difference equation, logistic equation, series summation, transseries

Identifiers

Local EPrints ID: 452982
URI: http://eprints.soton.ac.uk/id/eprint/452982
ISSN: 0951-7715
PURE UUID: f156062e-e376-4b0e-9f27-7d35136adc6c
ORCID for Ines Aniceto: ORCID iD orcid.org/0000-0002-4468-0066
ORCID for Christopher Howls: ORCID iD orcid.org/0000-0001-7989-7807

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Date deposited: 07 Jan 2022 12:09
Last modified: 17 Mar 2024 03:54

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Contributors

Author: Ines Aniceto ORCID iD
Author: Daniel Hasenbichler
Author: Christopher Lustri

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