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

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.

💡 Research Summary

This paper addresses the challenging problem of constructing heteroclinic connections between two distinct hyperbolic periodic orbits (PtoP fronts) in conservative dynamical systems. While heteroclinic orbits involving equilibria (EtoE) or equilibrium–periodic pairs (EtoP) are relatively well understood, connections between periodic orbits have received far less attention due to the difficulty of locating large‑amplitude solutions and computing them accurately. The authors propose a unified theoretical and numerical framework that uses a codimension‑two singularity in a family of periodic orbits as an organizing center for the emergence of heteroclinic connections.

Theoretical contribution:

1. Normal‑form derivation – By performing a phase‑modulation expansion around a family of periodic solutions, the authors derive a nonlinear normal form that captures the interaction of two branches of periodic orbits near a codimension‑two singularity. The normal form takes the simple form
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