High-order adaptive method for computing two-dimensional invariant manifolds of three-dimensional maps

Jacek K. Wróbel; Roy H. Goodman

doi:10.1016/j.cnsns.2012.10.017

High-order adaptive method for computing two-dimensional invariant manifolds of three-dimensional maps

Jacek K. Wróbel, Roy H. Goodman

Mathematical Sciences

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

12 Scopus citations

Abstract

An efficient and accurate numerical method is presented for computing invariant manifolds of maps which arise in the study of dynamical systems. A quasi-interpolation method due to Hering-Bertram et al. is used to decrease the number of points needed to compute a portion of the manifold. Bézier triangular patches are used in this construction, together with adaptivity conditions based on properties of these patches. Several numerical tests are performed, which show the method to compare favorably with previous approaches.

Original language	English (US)
Pages (from-to)	1734-1745
Number of pages	12
Journal	Communications in Nonlinear Science and Numerical Simulation
Volume	18
Issue number	7
DOIs	https://doi.org/10.1016/j.cnsns.2012.10.017
State	Published - Jul 2013

All Science Journal Classification (ASJC) codes

Numerical Analysis
Modeling and Simulation
Applied Mathematics

Keywords

Computer-aided geometric design
Invariant manifold
Numerical algorithm

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10.1016/j.cnsns.2012.10.017

Cite this

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title = "High-order adaptive method for computing two-dimensional invariant manifolds of three-dimensional maps",

abstract = "An efficient and accurate numerical method is presented for computing invariant manifolds of maps which arise in the study of dynamical systems. A quasi-interpolation method due to Hering-Bertram et al. is used to decrease the number of points needed to compute a portion of the manifold. B{\'e}zier triangular patches are used in this construction, together with adaptivity conditions based on properties of these patches. Several numerical tests are performed, which show the method to compare favorably with previous approaches.",

keywords = "Computer-aided geometric design, Invariant manifold, Numerical algorithm",

author = "Wr{\'o}bel, {Jacek K.} and Goodman, {Roy H.}",

note = "Funding Information: Thanks to Gershon Elber, Rafael de la Llave, Hector Lomel{\'i}, James Meiss, Jason Mireles-James, and Denis Zorin for helpful discussions, to the anonymous referees, whose suggestions improved this paper enormously, and to Casayndra Basarab for her careful reading of the manuscript. The authors were supported by the NSF under Grant Nos. DMS-0807284 , DMS-0639270. Copyright: Copyright 2013 Elsevier B.V., All rights reserved.",

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doi = "10.1016/j.cnsns.2012.10.017",

language = "English (US)",

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AU - Wróbel, Jacek K.

AU - Goodman, Roy H.

N1 - Funding Information: Thanks to Gershon Elber, Rafael de la Llave, Hector Lomelí, James Meiss, Jason Mireles-James, and Denis Zorin for helpful discussions, to the anonymous referees, whose suggestions improved this paper enormously, and to Casayndra Basarab for her careful reading of the manuscript. The authors were supported by the NSF under Grant Nos. DMS-0807284 , DMS-0639270. Copyright: Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2013/7

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N2 - An efficient and accurate numerical method is presented for computing invariant manifolds of maps which arise in the study of dynamical systems. A quasi-interpolation method due to Hering-Bertram et al. is used to decrease the number of points needed to compute a portion of the manifold. Bézier triangular patches are used in this construction, together with adaptivity conditions based on properties of these patches. Several numerical tests are performed, which show the method to compare favorably with previous approaches.

AB - An efficient and accurate numerical method is presented for computing invariant manifolds of maps which arise in the study of dynamical systems. A quasi-interpolation method due to Hering-Bertram et al. is used to decrease the number of points needed to compute a portion of the manifold. Bézier triangular patches are used in this construction, together with adaptivity conditions based on properties of these patches. Several numerical tests are performed, which show the method to compare favorably with previous approaches.

KW - Computer-aided geometric design

KW - Invariant manifold

KW - Numerical algorithm

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M3 - Article

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JO - Communications in Nonlinear Science and Numerical Simulation

JF - Communications in Nonlinear Science and Numerical Simulation

SN - 1007-5704

IS - 7

ER -

High-order adaptive method for computing two-dimensional invariant manifolds of three-dimensional maps

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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