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

TL;DR

This paper investigates heteroclinic connections between finite-amplitude periodic orbits emerging from a codimension two singularity, combining normal form theory and numerical continuation to analyze their properties in conservative systems.

Contribution

It introduces a normal form for the emergence of heteroclinic connections from a codimension two singularity and develops numerical strategies to compute these connections far from the singularity.

Findings

Derived a normal form capturing heteroclinic connections from a singularity.

Developed numerical continuation methods using shooting and core-farfield decomposition.

Applied theory to equations in fluid dynamics and pattern formation.

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.