Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation

Christian Reinhardt, J. D. Mireles James

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We study polynomial expansions of local unstable manifolds attached to equilibrium solutions of parabolic partial differential equations. Due to the smoothing properties of parabolic equations, these manifolds are finite dimensional. Our approach is based on an infinitesimal invariance equation and recovers the dynamics on the manifold in addition to its embedding. The invariance equation is solved, to any desired order in space and time, using a Newton scheme on the space of formal Fourier–Taylor series. Under mild non-resonance conditions we show that the formal series converge in some small enough neighborhood of the equilibrium. An a-posteriori computer assisted argument is given which, when successful, provides mathematically rigorous convergence proofs in explicit and much larger neighborhoods. We give example computations and applications.

Original languageEnglish
Pages (from-to)39-80
Number of pages42
JournalIndagationes Mathematicae
Volume30
Issue number1
Early online date23 Aug 2018
DOIs
Publication statusPublished - Jan 2019

Fingerprint

Unstable Manifold
Parabolic Partial Differential Equations
Parameterization
Invariance
Nonresonance
Series
Equilibrium Solution
Parabolic Equation
Smoothing
Converge
Polynomial

Keywords

  • Computer-assisted proof
  • Connecting orbit
  • Parabolic partial differential equations
  • Parameterization method
  • Unstable manifold

Cite this

@article{8c6cdad990d54f45a15f759d380e555f,
title = "Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation",
abstract = "We study polynomial expansions of local unstable manifolds attached to equilibrium solutions of parabolic partial differential equations. Due to the smoothing properties of parabolic equations, these manifolds are finite dimensional. Our approach is based on an infinitesimal invariance equation and recovers the dynamics on the manifold in addition to its embedding. The invariance equation is solved, to any desired order in space and time, using a Newton scheme on the space of formal Fourier–Taylor series. Under mild non-resonance conditions we show that the formal series converge in some small enough neighborhood of the equilibrium. An a-posteriori computer assisted argument is given which, when successful, provides mathematically rigorous convergence proofs in explicit and much larger neighborhoods. We give example computations and applications.",
keywords = "Computer-assisted proof, Connecting orbit, Parabolic partial differential equations, Parameterization method, Unstable manifold",
author = "Christian Reinhardt and {Mireles James}, {J. D.}",
year = "2019",
month = "1",
doi = "10.1016/j.indag.2018.08.003",
language = "English",
volume = "30",
pages = "39--80",
journal = "Indagationes Mathematicae",
issn = "0019-3577",
publisher = "Elsevier",
number = "1",

}

Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations : Formalism, implementation and rigorous validation. / Reinhardt, Christian; Mireles James, J. D.

In: Indagationes Mathematicae, Vol. 30, No. 1, 01.2019, p. 39-80.

Research output: Contribution to JournalArticleAcademicpeer-review

TY - JOUR

T1 - Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations

T2 - Formalism, implementation and rigorous validation

AU - Reinhardt, Christian

AU - Mireles James, J. D.

PY - 2019/1

Y1 - 2019/1

N2 - We study polynomial expansions of local unstable manifolds attached to equilibrium solutions of parabolic partial differential equations. Due to the smoothing properties of parabolic equations, these manifolds are finite dimensional. Our approach is based on an infinitesimal invariance equation and recovers the dynamics on the manifold in addition to its embedding. The invariance equation is solved, to any desired order in space and time, using a Newton scheme on the space of formal Fourier–Taylor series. Under mild non-resonance conditions we show that the formal series converge in some small enough neighborhood of the equilibrium. An a-posteriori computer assisted argument is given which, when successful, provides mathematically rigorous convergence proofs in explicit and much larger neighborhoods. We give example computations and applications.

AB - We study polynomial expansions of local unstable manifolds attached to equilibrium solutions of parabolic partial differential equations. Due to the smoothing properties of parabolic equations, these manifolds are finite dimensional. Our approach is based on an infinitesimal invariance equation and recovers the dynamics on the manifold in addition to its embedding. The invariance equation is solved, to any desired order in space and time, using a Newton scheme on the space of formal Fourier–Taylor series. Under mild non-resonance conditions we show that the formal series converge in some small enough neighborhood of the equilibrium. An a-posteriori computer assisted argument is given which, when successful, provides mathematically rigorous convergence proofs in explicit and much larger neighborhoods. We give example computations and applications.

KW - Computer-assisted proof

KW - Connecting orbit

KW - Parabolic partial differential equations

KW - Parameterization method

KW - Unstable manifold

UR - http://www.scopus.com/inward/record.url?scp=85053666261&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85053666261&partnerID=8YFLogxK

U2 - 10.1016/j.indag.2018.08.003

DO - 10.1016/j.indag.2018.08.003

M3 - Article

VL - 30

SP - 39

EP - 80

JO - Indagationes Mathematicae

JF - Indagationes Mathematicae

SN - 0019-3577

IS - 1

ER -