Rigorous Numerics for Global Dynamics: A Study of the Swift--Hohenberg Equation | Zendy

Sarah Day | Zendy; Yasuaki Hiraoka | Zendy; Konstantin Mischaikow | Zendy; Toshiyuki OGAWA | Zendy

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Rigorous Numerics for Global Dynamics: A Study of the Swift--Hohenberg Equation

Author(s) -

Sarah Day,

Yasuaki Hiraoka,

Konstantin Mischaikow,

Toshiyuki OGAWA

Publication year - 2005

Publication title -

siam journal on applied dynamical systems

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.218

H-Index - 61

ISSN - 1536-0040

DOI - 10.1137/040604479

Subject(s) - attractor , uniqueness , swift , continuation , bifurcation , mathematics , bifurcation diagram , topological conjugacy , conjugacy class , numerical continuation , mathematical analysis , computer science , pure mathematics , nonlinear system , physics , programming language , quantum mechanics

This paper presents a rigorous numerical method for the study and verifi- cation of global dynamics. In particular, this method produces a conjugacy or semi-conjugacy 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 con- necting orbits.

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