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Heteroclinic for a 6-dimensional reversible system occuring in orthogonal domain walls in convection

Gérard Iooss

TL;DR

The paper proves the existence, local uniqueness, and analyticity of a heteroclinic connection between orthogonal-convection roll states in a six-dimensional reversible normal-form system derived from Benard-Rayleigh convection. By constructing a 5D invariant manifold on which a 3D unstable manifold from M_- intersects a 3D stable manifold from M_+, the authors establish transversal intersection and derive precise asymptotics for the heteroclinic orbit, with the heteroclinic shown to depend analytically on the parameters (notably g). A detailed linearized analysis along the heteroclinic yields Fredholm properties and a one-dimensional kernel, enabling persistence results under reversible perturbations and linking to orthogonal domain walls in convection. The approach hinges on careful coordinate changes, a first-integral constraint, integral formulations, and fixed-point arguments, culminating in a rigorous, parameter-analytic heteroclinic curve and a framework for studying perturbations. These results provide a rigorous foundation for domain-wall persistence and offer sharp estimates essential for applications to convection phenomena.

Abstract

A six-dimensional reversible normal form system occurs in B{é}nard-Rayleigh convection between parallel planes, when we look for domain walls intersecting orthogonally (see Buffoni et al [1]). On the truncated system, we prove analytically the existence, local uniqueness, and analyticity in parameters, of a heteroclinic connection between two equilibria, each corresponding to a system of convective rolls. We prove that the 3-dimensional unstable manifold of one equilibrium, intersects transversally the 3-dimensional stable manifold of the other equilibrium, both manifolds lying on a 5-dimensional invariant manifold. We also study the linearized operator along the heteroclinic, allowing to prove (in [9]) the persistence under reversible perturbation, of the heteroclinic obtained in [1].

Heteroclinic for a 6-dimensional reversible system occuring in orthogonal domain walls in convection

TL;DR

The paper proves the existence, local uniqueness, and analyticity of a heteroclinic connection between orthogonal-convection roll states in a six-dimensional reversible normal-form system derived from Benard-Rayleigh convection. By constructing a 5D invariant manifold on which a 3D unstable manifold from M_- intersects a 3D stable manifold from M_+, the authors establish transversal intersection and derive precise asymptotics for the heteroclinic orbit, with the heteroclinic shown to depend analytically on the parameters (notably g). A detailed linearized analysis along the heteroclinic yields Fredholm properties and a one-dimensional kernel, enabling persistence results under reversible perturbations and linking to orthogonal domain walls in convection. The approach hinges on careful coordinate changes, a first-integral constraint, integral formulations, and fixed-point arguments, culminating in a rigorous, parameter-analytic heteroclinic curve and a framework for studying perturbations. These results provide a rigorous foundation for domain-wall persistence and offer sharp estimates essential for applications to convection phenomena.

Abstract

A six-dimensional reversible normal form system occurs in B{é}nard-Rayleigh convection between parallel planes, when we look for domain walls intersecting orthogonally (see Buffoni et al [1]). On the truncated system, we prove analytically the existence, local uniqueness, and analyticity in parameters, of a heteroclinic connection between two equilibria, each corresponding to a system of convective rolls. We prove that the 3-dimensional unstable manifold of one equilibrium, intersects transversally the 3-dimensional stable manifold of the other equilibrium, both manifolds lying on a 5-dimensional invariant manifold. We also study the linearized operator along the heteroclinic, allowing to prove (in [9]) the persistence under reversible perturbation, of the heteroclinic obtained in [1].

Paper Structure

This paper contains 46 sections, 19 theorems, 351 equations, 2 figures.

Table of Contents

  1. Introduction and Results
  2. Origin of system (\ref{['truncated syst']})
  3. Sketch of the method and results
  4. General properties of the system
  5. Global invariant manifold $\mathcal{W}_{{ \if@compatibility \mathchar"0122 {} \mathchar"0122 } ,{ \if@compatibility \mathchar"010E {} \mathchar"010E } }$
  6. Linear study of the dynamics
  7. Neighborhood of $M_{-}=(1,0)$
  8. Neighborhood of $M_{+}=(0,1)$
  9. Unstable manifold of $M_{\_}$
  10. Choice of coordinates
  11. New coordinates
  12. System with new coordinates
  13. Resolution of (\ref{['Wg']}) with respect of $z_{1}(X,Y,B_{0})$
  14. System where $z_{1}$ is eliminated
  15. Integral formulation for solutions bounded as $x\rightarrow -\infty$
  16. ...and 31 more sections

Key Result

Theorem 1

Let us choose $1/3\leq { \if@compatibility \mathchar"010E {} \mathchar"010E } \leq 1$, then for ${ \if@compatibility \mathchar"0122 {} \mathchar"0122 }$ small enough, the 3-dim unstable manifold of $M_{-}$ intersects transversally the 3-dim stable manifold of $M_{+},$ except maybe for a

Figures (2)

  • Figure 1: Critical manifold
  • Figure 2: Orthogonal domain wall

Theorems & Definitions (42)

  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Remark 7
  • ...and 32 more