Rigorous numerics for global dynamics: A study of the swift-hohenberg equation

Sarah Day; Yasuaki Hiraoka; Konstantin Mischaikow; Toshiyuki Ogawa

doi:10.1137/040604479

Rigorous numerics for global dynamics: A study of the swift-hohenberg equation

Sarah Day, Yasuaki Hiraoka, Konstantin Mischaikow, Toshiyuki Ogawa

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

43 Citations (Scopus)

Abstract

This paper presents a rigorous numerical method for the study and verification of global dynamics. In particular, this method produces a conjugacy or semiconjugacy between an attractor for the Swift-Hohenberg equation and a model system. The procedure involved relies on first verifying bifurcation diagrams produced via continuation methods, including proving the existence and uniqueness of computed branches as well as showing the nonexistence of additional stationary solutions. Topological information in the form of the Conley index, also computed during this verification procedure, is then used to build a model for the attractor consisting of stationary solutions and connecting orbits.

Original language	English
Pages (from-to)	1-31
Number of pages	31
Journal	SIAM Journal on Applied Dynamical Systems
Volume	4
Issue number	1
DOIs	https://doi.org/10.1137/040604479
Publication status	Published - 2005
Externally published	Yes

Keywords

Conley index
Rigorous numerics
Semiconjugacy

ASJC Scopus subject areas

Analysis
Modelling and Simulation

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abstract = "This paper presents a rigorous numerical method for the study and verification of global dynamics. In particular, this method produces a conjugacy or semiconjugacy between an attractor for the Swift-Hohenberg equation and a model system. The procedure involved relies on first verifying bifurcation diagrams produced via continuation methods, including proving the existence and uniqueness of computed branches as well as showing the nonexistence of additional stationary solutions. Topological information in the form of the Conley index, also computed during this verification procedure, is then used to build a model for the attractor consisting of stationary solutions and connecting orbits.",

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AB - This paper presents a rigorous numerical method for the study and verification of global dynamics. In particular, this method produces a conjugacy or semiconjugacy between an attractor for the Swift-Hohenberg equation and a model system. The procedure involved relies on first verifying bifurcation diagrams produced via continuation methods, including proving the existence and uniqueness of computed branches as well as showing the nonexistence of additional stationary solutions. Topological information in the form of the Conley index, also computed during this verification procedure, is then used to build a model for the attractor consisting of stationary solutions and connecting orbits.

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KW - Semiconjugacy

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Rigorous numerics for global dynamics: A study of the swift-hohenberg equation

Abstract

Keywords

ASJC Scopus subject areas

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