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

Advanced Institute for Materials Research (AIMR)

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

36 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

Keywords

Conley index
Rigorous numerics
Semiconjugacy

ASJC Scopus subject areas

Analysis
Modelling and Simulation

Access to Document

10.1137/040604479

Fingerprint Dive into the research topics of 'Rigorous numerics for global dynamics: A study of the swift-hohenberg equation'. Together they form a unique fingerprint.

Cite this

@article{3ee2239a200f4ea89bbf1fa691965799,

title = "Rigorous numerics for global dynamics: A study of the swift-hohenberg equation",

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.",

keywords = "Conley index, Rigorous numerics, Semiconjugacy",

author = "Sarah Day and Yasuaki Hiraoka and Konstantin Mischaikow and Toshiyuki Ogawa",

year = "2005",

doi = "10.1137/040604479",

language = "English",

volume = "4",

pages = "1--31",

journal = "SIAM Journal on Applied Dynamical Systems",

issn = "1536-0040",

publisher = "Society of Industrial and Applied Mathematics",

number = "1",

}

TY - JOUR

T1 - Rigorous numerics for global dynamics

T2 - A study of the swift-hohenberg equation

AU - Day, Sarah

AU - Hiraoka, Yasuaki

AU - Mischaikow, Konstantin

AU - Ogawa, Toshiyuki

PY - 2005

Y1 - 2005

N2 - 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.

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.

KW - Conley index

KW - Rigorous numerics

KW - Semiconjugacy

UR - http://www.scopus.com/inward/record.url?scp=33644899149&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33644899149&partnerID=8YFLogxK

U2 - 10.1137/040604479

DO - 10.1137/040604479

M3 - Article

AN - SCOPUS:33644899149

VL - 4

SP - 1

EP - 31

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 1

ER -