Continuation sheaves in dynamics: Sheaf cohomology and bifurcation

K. Alex Dowling; William D. Kalies; Robert C.A.M. Vandervorst

doi:10.1016/j.jde.2023.04.041

Continuation sheaves in dynamics: Sheaf cohomology and bifurcation

K. Alex Dowling^*, William D. Kalies, Robert C.A.M. Vandervorst

^*Corresponding author for this work

Mathematics

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Abstract

Algebraic structures such as the lattices of attractors, repellers, and Morse representations provide a computable description of global dynamics. In this paper, a sheaf-theoretic approach to their continuation is developed. The algebraic structures are cast into a categorical framework to study their continuation systematically and simultaneously. Sheaves are built from this abstract formulation, which track the algebraic data as systems vary. Sheaf cohomology is computed for several classical bifurcations, demonstrating its ability to detect and classify bifurcations.

Original language	English
Pages (from-to)	124-198
Number of pages	75
Journal	Journal of Differential Equations
Volume	367
Early online date	12 May 2023
DOIs	https://doi.org/10.1016/j.jde.2023.04.041
Publication status	Published - 15 Sept 2023

Bibliographical note

Funding Information:
The work of W.D.K. was partially supported by the Army Research Office under award W911NF1810306.

Publisher Copyright:
© 2023 Elsevier Inc.

Funding

The work of W.D.K. was partially supported by the Army Research Office under award W911NF1810306.

Keywords

Attractor sheaf
Bifurcation
Category of dynamical systems
Continuation
Morse representation/decomposition
Sheaf cohomology

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10.1016/j.jde.2023.04.041

Continuation sheaves in dynamics_Sheaf cohomology and bifurcationFinal published version, 2.21 MBLicence: Article 25fa Dutch Copyright Act

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AU - Kalies, William D.

AU - Vandervorst, Robert C.A.M.

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Y1 - 2023/9/15

N2 - Algebraic structures such as the lattices of attractors, repellers, and Morse representations provide a computable description of global dynamics. In this paper, a sheaf-theoretic approach to their continuation is developed. The algebraic structures are cast into a categorical framework to study their continuation systematically and simultaneously. Sheaves are built from this abstract formulation, which track the algebraic data as systems vary. Sheaf cohomology is computed for several classical bifurcations, demonstrating its ability to detect and classify bifurcations.

AB - Algebraic structures such as the lattices of attractors, repellers, and Morse representations provide a computable description of global dynamics. In this paper, a sheaf-theoretic approach to their continuation is developed. The algebraic structures are cast into a categorical framework to study their continuation systematically and simultaneously. Sheaves are built from this abstract formulation, which track the algebraic data as systems vary. Sheaf cohomology is computed for several classical bifurcations, demonstrating its ability to detect and classify bifurcations.

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KW - Bifurcation

KW - Category of dynamical systems

KW - Continuation

KW - Morse representation/decomposition

KW - Sheaf cohomology

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SP - 124

EP - 198

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -

Continuation sheaves in dynamics: Sheaf cohomology and bifurcation

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