Generic Lyapunov instability of elliptic equilibria in many degrees of freedom
In plain words
At an equilibrium where all nearby motions oscillate, orbits that start very close may still drift away very slowly. This is expected for typical systems with three or more degrees of freedom, but it is proven only for special constructions and for five or more degrees of freedom.
Precise statement
Real-analytic $H$ on $R^{2n}$, $n \ge 3$, with $H = \operatorname{sum}_i \omega_i (p_i^2 + q_i^2)/2 + O(3)$, $\omega$ non-resonant and not all $\omega_i$ of one sign. Prove that the origin is Lyapunov unstable for generic higher-order terms (open dense or residual set in a suitable analytic topology). Answer: a proof, or a stable open set of examples.
What would settle it
A proof of generic Lyapunov instability for some $n \ge 3$ among all real-analytic Hamiltonians with given quadratic part.
Status in the literature
Unverified note
Fayad (arXiv 1809.09059, 2018) gave explicit real entire unstable examples in four or more degrees of freedom; Fayad, Paradela, Saprykina and Seara (arXiv 2609.03722, 2026) proved that for $n \ge 5$ any analytic Hamiltonian with a locally integrable non-degenerate elliptic equilibrium of indefinite quadratic part can be analytically perturbed to an unstable one; $n = 3, 4$ and full genericity remain open.