Arnold’s conjecture on generic Lyapunov instability

Determine whether, apart from the cases of two degrees of freedom and sign-definite quadratic part, elliptic equilibria of generic Hamiltonian systems are Lyapunov unstable.

Background

The paper studies Lyapunov stability near elliptic equilibria whose linearized dynamics is stable but whose nonlinear behavior can be affected by resonances and diffusion. Arnold conjectured that effective stability properties supplied by Birkhoff normal form and KAM theory do not generally imply Lyapunov stability.

The conjecture excludes two exceptional situations: Hamiltonians with sign-definite quadratic part, for which energy preservation prevents local Lyapunov instability, and systems with two degrees of freedom. The paper proves a density result in the real-analytic category for systems with at least five degrees of freedom, thereby providing partial progress but not resolving the full genericity conjecture.

References

Despite these strong stability properties, Arnold conjectured that they do not, in general, imply Lyapunov stability. More precisely, he conjectured that, apart from the two exceptional situations where the quadratic part of the Hamiltonian is sign-definite (equivalently, all frequencies have the same sign) and the two degrees of freedom case, elliptic equilibria of generic Hamiltonian systems should be Lyapunov unstable. The latter conjecture remains one of the central open problems in Hamiltonian dynamics and motivates the present work.

— On the density of Lyapunov unstable elliptic equilibria  (2609.03722 - Fayad et al., 3 Sep 2026) in Section 1, Introduction and main results

This issue is closely related to the conjectural density of locally integrable real analytic Hamiltonians, a problem for which a proof has recently been announced by Krikorian.

— On the density of Lyapunov unstable elliptic equilibria  (2609.03722 - Fayad et al., 3 Sep 2026) in Section 1, Introduction and main results; Section 2, subsection “Density of Lyapunov unstable equilibria”