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Heteroclinic connections between finite-amplitude periodic orbits emerging from a codimension two singularity

Thomas J. Bridges, David J. B. Lloyd, Daniel J. Ratliff, Patrick Sprenger

TL;DR

The paper develops a geometric and computational framework for heteroclinic connections between finite-amplitude periodic orbits in conservative systems, centering on a codimension-two singularity in the wavenumber family and an unfolding normal form that yields explicit heteroclinic fronts. It introduces action as a central invariant along stable and unstable foliations, establishes its relation to the Hamiltonian and Bloch spectrum, and shows a characteristic jump in action across a surface of section. A robust predictor–corrector numerical strategy (shooting for foliations and core–farfield decomposition for refinement) is demonstrated on the Swift–Hohenberg family (including SH357), generalized NLS with higher-order dispersion, and a coupled KdV/Boussinesq system, producing both symmetry-related and symmetry-broken PtoP fronts. The results offer a universal mechanism and practical computation method for predicting and continuing heteroclinic fronts in dispersive pattern-forming systems, with potential extensions to higher-dimensional PDEs and multi-phonon invariant structures.

Abstract

Heteroclinic connections between two distinct hyperbolic periodic orbits in conservative systems are important in a wide range of applications. On the other hand, it is theoretically challenging to find large amplitude connections from scratch and compute them numerically. In this paper, we use a codimension two singularity, in a family of periodic orbits, as an organizing center for the emergence of heteroclinic connections. A normal form is derived whose unfolding produces two distinct finite amplitude periodic orbits with an explicit heteroclinic connection. We also construct heteroclinic connections far from the singularity by numerical continuation, using two numerical strategies: shooting and the core-farfield decomposition. A key geometric tool in the numerics is cylindrical foliations for the stable and unstable manifolds and their intersection. We introduce a new property of heteroclinic connections - the action - and show it is an invariant along foliations, it has a jump at a surface of section, and it appears in a central way in the normal form theory. We find that the difference in asymptotic phase between minus and plus infinity is also a key property. The theory is applied to the Swift-Hohenberg equation, the nonlinear Schrodinger with fourth order dispersion, and coupled Boussinesq equations from water waves, all of which have an energy and action conservation law.

Heteroclinic connections between finite-amplitude periodic orbits emerging from a codimension two singularity

TL;DR

The paper develops a geometric and computational framework for heteroclinic connections between finite-amplitude periodic orbits in conservative systems, centering on a codimension-two singularity in the wavenumber family and an unfolding normal form that yields explicit heteroclinic fronts. It introduces action as a central invariant along stable and unstable foliations, establishes its relation to the Hamiltonian and Bloch spectrum, and shows a characteristic jump in action across a surface of section. A robust predictor–corrector numerical strategy (shooting for foliations and core–farfield decomposition for refinement) is demonstrated on the Swift–Hohenberg family (including SH357), generalized NLS with higher-order dispersion, and a coupled KdV/Boussinesq system, producing both symmetry-related and symmetry-broken PtoP fronts. The results offer a universal mechanism and practical computation method for predicting and continuing heteroclinic fronts in dispersive pattern-forming systems, with potential extensions to higher-dimensional PDEs and multi-phonon invariant structures.

Abstract

Heteroclinic connections between two distinct hyperbolic periodic orbits in conservative systems are important in a wide range of applications. On the other hand, it is theoretically challenging to find large amplitude connections from scratch and compute them numerically. In this paper, we use a codimension two singularity, in a family of periodic orbits, as an organizing center for the emergence of heteroclinic connections. A normal form is derived whose unfolding produces two distinct finite amplitude periodic orbits with an explicit heteroclinic connection. We also construct heteroclinic connections far from the singularity by numerical continuation, using two numerical strategies: shooting and the core-farfield decomposition. A key geometric tool in the numerics is cylindrical foliations for the stable and unstable manifolds and their intersection. We introduce a new property of heteroclinic connections - the action - and show it is an invariant along foliations, it has a jump at a surface of section, and it appears in a central way in the normal form theory. We find that the difference in asymptotic phase between minus and plus infinity is also a key property. The theory is applied to the Swift-Hohenberg equation, the nonlinear Schrodinger with fourth order dispersion, and coupled Boussinesq equations from water waves, all of which have an energy and action conservation law.
Paper Structure (19 sections, 4 theorems, 176 equations, 17 figures)

This paper contains 19 sections, 4 theorems, 176 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Computing foliations in the Generalized NLS Equation
  3. Connecting symmetry-related periodic orbits
  4. Connecting asymmetric periodic orbits
  5. Symmetry breaking
  6. Properties of periodic orbits
  7. Action and algebraic multiplicity
  8. Nonlinear normal form theory
  9. Heteroclinic connections via the normal form
  10. The SH357 equation
  11. Finding and continuing codimension two points
  12. Heteroclinic connections near codimension two points
  13. Fronts near codimension-two points in a Boussinesq system
  14. Concluding remarks
  15. Derivatives of the action
  16. ...and 4 more sections

Key Result

Lemma 3.1

For the equation (primary-ode) with Hamiltonian $H$ in (H-def) and action (A-per) evaluated on a family of periodic orbits with $k\neq0$,

Figures (17)

  • Figure 1: Examples of a heteroclinic connection between two distinct periodic states (details of this example are in §\ref{['sec-SH357-review']}). Panel (a) shows a heteroclinic connection between two periodic orbits in the cubic SH equation with $\sigma=1$ and panel (b) shows a heteroclinic connection in the SH357 equation with $\mu = 0.305$, $a = 1.438$, and $b = 2.117$.
  • Figure 2: Schematic of the computation of the stable and unstable foliations: panel (a) shows a heteroclinic connection associated with one of the intersection points in (c); panel (b) shows cylindrical foliations obtained by solving the inverse problem for the asymptotic phase; and panel (c) shows the surface of section at $x=0$ illustrating the intersection between the stable (blue) and unstable (red) foliations.
  • Figure 3: Plot of the value of the Hamiltonian and the Action function versus the wavenumber along a branch of periodic states near an inflection point. The solid curve corresponds to the spatial Hamiltonian and the dashed curve to the Action. Arrows indicate the motion of the critical points as the codimension 2 point is approached.
  • Figure 4: A family of periodic solutions of (\ref{['per-eqn-nls']}) with $\sigma=1$. In panel (a) the Hamitonian and Action are plotted as functions of $k$. Although $H$ and $A$ are qualitatively different, their critical points agree. Panels (b), (c) and (d) show plots of the periodic solutions $\widehat{u}(z,k)$ as a function of $z$, with $-\pi < z < +\pi$, at three $k-$points along the curves.
  • Figure 5: A plot of the stable and unstable foliations of the periodic orbits in \ref{['first-order-numerics']}. Panel (a) shows the stable and unstable manifolds (colored blue and red respectively) starting from the periodic orbits and ending on the (colored yellow) Poincaré section. Panel (b) shows a plot of the Poincaré section, with the two intersection points colored black. Panel (c) includes the manifolds along with the highlighted leaf which is the connection. That connection is shown as a function of $x$ in panel (d). Panel (e) Shows a plot of the other heteroclinic connection on the manifolds and corresponding heteroclinic connection as a function of $x$ is in (f)
  • ...and 12 more figures

Theorems & Definitions (13)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Proposition 3.4
  • Proposition 3.5
  • proof
  • Remark 3.6
  • Remark 4.1
  • Lemma 5.1
  • ...and 3 more