Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4

Andrew Burbanks; Andrew Osbaldestin; Judi Thurlby

doi:10.1063/5.0054823

Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4

Andrew Burbanks^*, Andrew Osbaldestin, Judi Thurlby

^*Corresponding author for this work

School of Mathematics & Physics

Research output: Contribution to journal › Article › peer-review

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Abstract

We gain tight rigorous bounds on the renormalisation fixed point for period doubling in families of unimodal maps with degree 4 critical point. We use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearisation of the operator and for the operator controlling the scaling of added noise. Multi-precision arithmetic with rigorous directed rounding is used to bound operations in a space of analytic functions yielding tight bounds on power series coefficients and universal constants to over 320 significant figures.

Original language	English
Article number	112701
Number of pages	18
Journal	Journal of Mathematical Physics
Volume	62
Issue number	11
Early online date	3 Nov 2021
DOIs	https://doi.org/10.1063/5.0054823
Publication status	Published - 30 Nov 2021

Keywords

dynamical systems
renormalisation group
universality
period-doubling
bifurcations
computer-assisted proofs

Access to Document

10.1063/5.0054823

JMP21-AR-00685
This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in J. Math. Phys. 62, 112701 (2021) and may be found at https://doi.org/10.1063/5.0054823.
Accepted author manuscript (Post-print), 1.1 MB

2 Citations
1 Article

Existence of the FS-type renormalisation ﬁxed point for unidirectionally-coupled pairs of maps
Burbanks, A. D. & Osbaldestin, A. H., 20 Apr 2023, In: Journal of Physics A: Mathematical and Theoretical. 56, 17 p., 195202.
Research output: Contribution to journal › Article › peer-review

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Dataset for "Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4"
Burbanks, A. (Creator), Zenodo, 28 Oct 2021
DOI: 10.5281/zenodo.5608449
Dataset

2 Participation in conference
1 Invited talk

Invited speaker at MIRaW – Mathematical Interdisciplinary Research at Warwick event on "Computation in Mathematics and Mathematics in Computation"
Burbanks, A. (Invited speaker)
11 Mar 2024
Activity: Talk or presentation types › Invited talk
JuliaCon 2023
Burbanks, A. (Participant)
25 Jul 2023 → 29 Jul 2023
Activity: Participating in or organising an event types › Participation in conference
British Applied Mathematics Colloquium (BAMC) 2023
Burbanks, A. (Presented paper)
3 Apr 2023 → 5 Apr 2023
Activity: Participating in or organising an event types › Participation in conference

Cite this

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title = "Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4",

abstract = "We gain tight rigorous bounds on the renormalisation fixed point for period doubling in families of unimodal maps with degree 4 critical point. We use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearisation of the operator and for the operator controlling the scaling of added noise. Multi-precision arithmetic with rigorous directed rounding is used to bound operations in a space of analytic functions yielding tight bounds on power series coefficients and universal constants to over 320 significant figures.",

keywords = "dynamical systems, renormalisation group, universality, period-doubling, bifurcations, computer-assisted proofs",

author = "Andrew Burbanks and Andrew Osbaldestin and Judi Thurlby",

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AU - Burbanks, Andrew

AU - Osbaldestin, Andrew

AU - Thurlby, Judi

N1 - No embargo. Publisher's Version of Record can be made OA after a 12 month embargo.

PY - 2021/11/30

Y1 - 2021/11/30

N2 - We gain tight rigorous bounds on the renormalisation fixed point for period doubling in families of unimodal maps with degree 4 critical point. We use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearisation of the operator and for the operator controlling the scaling of added noise. Multi-precision arithmetic with rigorous directed rounding is used to bound operations in a space of analytic functions yielding tight bounds on power series coefficients and universal constants to over 320 significant figures.

AB - We gain tight rigorous bounds on the renormalisation fixed point for period doubling in families of unimodal maps with degree 4 critical point. We use a contraction mapping argument to bound essential eigenfunctions and eigenvalues for the linearisation of the operator and for the operator controlling the scaling of added noise. Multi-precision arithmetic with rigorous directed rounding is used to bound operations in a space of analytic functions yielding tight bounds on power series coefficients and universal constants to over 320 significant figures.

KW - dynamical systems

KW - renormalisation group

KW - universality

KW - period-doubling

KW - bifurcations

KW - computer-assisted proofs

UR - https://doi.org/10.5281/zenodo.5608449

U2 - 10.1063/5.0054823

DO - 10.1063/5.0054823

M3 - Article

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VL - 62

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 11

M1 - 112701

ER -

Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4

Abstract

Keywords

Access to Document

Fingerprint

Research output

Existence of the FS-type renormalisation ﬁxed point for unidirectionally-coupled pairs of maps

Datasets

Dataset for "Rigorous computer-assisted bounds on the period doubling renormalisation fixed point and eigenfunctions in maps with critical point of degree 4"

Activities

Invited speaker at MIRaW – Mathematical Interdisciplinary Research at Warwick event on "Computation in Mathematics and Mathematics in Computation"

JuliaCon 2023

British Applied Mathematics Colloquium (BAMC) 2023

Cite this