Lyapunov instability in KAM stable Hamiltonians with two degrees of freedom

TL;DR

This paper constructs Gevrey-smooth Hamiltonians close to integrable systems that exhibit Lyapunov unstable elliptic equilibria, yet remain KAM stable with abundant invariant tori, revealing complex stability phenomena in Hamiltonian dynamics.

Contribution

It demonstrates the existence of Gevrey-smooth Hamiltonians with Lyapunov unstable elliptic equilibria that are still KAM stable, near integrable systems with specific frequency conditions.

Findings

Existence of Lyapunov unstable elliptic equilibria near integrable Hamiltonians.

Construction of KAM stable equilibria with positive measure invariant tori.

Extension of results to neighborhoods of invariant tori with arbitrary rotation vectors.

Abstract

For a fixed frequency vector $\omega \in \mathbb{R}^2 \, \setminus \, \lbrace 0 \rbrace$ obeying $\omega_1 \omega_2 < 0$ we show the existence of Gevrey-smooth Hamiltonians, arbitrarily close to an integrable Kolmogorov non-degenerate analytic Hamiltonian, having a Lyapunov unstable elliptic equilibrium with frequency $\omega$. In particular, the elliptic fixed points thus constructed will be KAM stable, i.e. accumulated by invariant tori whose Lebesgue density tend to one in the neighbourhood of the point and whose frequencies cover a set of positive measure. Similar examples for near-integrable Hamiltonians in action-angle coordinates in the neighbourhood of a Lagragian invariant torus with arbitrary rotation vector are also given in this work.