Lyapunov unstable elliptic equilibria

TL;DR

This paper introduces a new diffusion mechanism near elliptic equilibria in Hamiltonian systems with three or more degrees of freedom, demonstrating Lyapunov instability with explicit examples and divergent Birkhoff normal forms.

Contribution

It provides explicit Hamiltonians with Lyapunov unstable elliptic equilibria and divergent Birkhoff normal forms for non-resonant frequencies in higher dimensions.

Findings

Explicit Hamiltonians with unstable equilibria in $ ^{2d}$ for $d extgreater 3$.

Construction of examples with divergent Birkhoff normal form.

Demonstration of instability for arbitrary non-resonant frequency vectors.

Abstract

A new diffusion mechanism from the neighborhood of elliptic equilibria for Hamiltonian flows in three or more degrees of freedom is introduced. We thus obtain explicit real entire Hamiltonians on $\R^{2d}$, $d\geq 4$, that have a Lyapunov unstable elliptic equilibrium with an arbitrary chosen frequency vector whose coordinates are not all of the same sign. For non-resonant frequency vectors, our examples all have divergent Birkhoff normal form at the equilibrium. On $\R^4$, we give explicit examples of real entire Hamiltonians having an equilibrium with an arbitrary chosen non-resonant frequency vector and a divergent Birkhoff normal form.