Lyapunov Unstable Motion Bifurcating from a Circular Vortex Filament

TL;DR

This paper constructs explicit solutions called axial screw motions that demonstrate the gap between orbital stability and Lyapunov instability in vortex filament dynamics.

Contribution

It proves the existence of bifurcating solutions near circular vortex filaments that are orbitally stable but Lyapunov unstable, clarifying stability distinctions.

Findings

Existence of axial screw motion solutions bifurcating from circular filaments.

Solutions remain close to the circle orbit but drift away over time.

Explicit examples of perturbations satisfying orbital stability but not Lyapunov stability.

Abstract

This paper investigates the dynamics of closed vortex filaments in $\R^3$ governed by the Localized Induction Equation. Recently, Aiki and Higaki (2026) established the nonlinear orbital stability of circular vortex filaments under asymmetric perturbations, while identifying Lyapunov instability due to the linear growth of translation modes. Motivated by this result, we prove the existence of a family of closed solutions, which we call axial screw motions, that bifurcate from a circular filament. These solutions remain uniformly close to the orbit of the circle, but drift secularly away from the reference motion because their translation speed along the symmetry axis differs from that of the circular filament. In particular, they provide explicit non-trivial perturbations that satisfy orbital-stability estimates while failing Lyapunov stability, thereby realizing the gap between orbital stability and Lyapunov stability near the circular filament.