For real-analytic Hamiltonian systems, the stability of elliptic equilibria and of quasi-periodic invariant tori is governed by the Birkhoff normal form (BNF), the formal power series in the actions obtained by removing the angle dependence of the Hamiltonian order by order. Recent constructions produced real entire Hamiltonians with Lyapunov-unstable quasi-periodic tori whose BNF converges, and real entire Hamiltonians with Lyapunov-unstable elliptic equilibria whose BNF diverges. What remains open, and is the central question of this problem, is the crossing of the two lines: does there exist a real-analytic Hamiltonian with a Lyapunov-unstable elliptic equilibrium whose BNF at the equilibrium converges? The same sources leave open whether Lyapunov-unstable elliptic equilibria exist at all in two degrees of freedom, or in three degrees of freedom with Diophantine frequency.

Background

An autonomous Hamiltonian system with dd degrees of freedom is given by a function HH on phase space R2d\mathbb{R}^{2d} (or Td×Rd\mathbb{T}^d\times\mathbb{R}^d in action-angle coordinates (θ,r)(\theta,r)) equipped with the canonical symplectic structure; its flow conserves HH. A Lagrangian torus on which the flow acts as a rigid rotation with constant velocity vector ω∈Rd\omega\in\mathbb{R}^d is a quasi-periodic (QP) torus of frequency ω\omega. A frequency vector is Diophantine if ∣⟨k,ω⟩∣≥γ∣k∣−τ|\langle k,\omega\rangle|\geq\gamma|k|^{-\tau} for some γ,τ>0\gamma,\tau>0 and all nonzero integer vectors kk — the small-divisor condition under which Kolmogorov-Arnold-Moser (KAM) theory guarantees that most invariant tori of an integrable system survive small perturbations — and Liouville if it fails every such condition. An equilibrium whose linearized flow has purely imaginary spectrum ±iω1,…,±iωd\pm i\omega_1,\dots,\pm i\omega_d with rationally independent ω\omega is a non-resonant elliptic equilibrium of frequency ω\omega. An invariant set is Lyapunov stable if all sufficiently nearby orbits remain close to it for all forward time, and Lyapunov unstable otherwise. When all coordinates of ω\omega have the same sign, the quadratic part of HH is definite and the classical Lagrange-Dirichlet criterion makes the equilibrium Lyapunov stable, so instability can only be sought for mixed-sign frequency vectors.

The Birkhoff normal form (BNF) at a QP torus or an elliptic equilibrium is the formal power series in the actions obtained by successively removing the angle dependence of HH order by order; it is always defined as a formal object at non-resonant frequencies, but the series may converge or diverge, and it is known to be generically divergent. A convergent normalizing transformation is equivalent to local analytic integrability, yet the BNF series itself may converge for non-integrable systems. What a convergent BNF implies for the dynamics is therefore a central question of stability theory; in the formulation attributed to Eliasson in the primary sources: if the system is non-integrable, can the BNF nevertheless converge?

For quasi-periodic tori this was settled recently in the analytic category. Farré-Fayad built, for any frequency ω∈Rd\omega\in\mathbb{R}^d, d≥3d\geq 3, real entire Hamiltonians — here meaning Hamiltonians holomorphic on the whole complexified phase space, the strongest form of analyticity — whose invariant QP torus is Lyapunov unstable while the BNF at the torus converges (it can even be chosen as an explicit polynomial), using an analytic version of the Anosov-Katok approximation-by-conjugations (AbC) method — a scheme in which a sequence of conjugates of an integrable system converges to a limit system with prescribed dynamics. For elliptic equilibria the frontier is set by Fayad: real entire Hamiltonians with Lyapunov-unstable elliptic equilibria exist for some non-resonant ω∈R3\omega\in\mathbb{R}^3 and for any mixed-sign ω∈Rd\omega\in\mathbb{R}^d with d≥4d\geq 4, but in all these non-resonant examples the BNF diverges. The equilibrium case is genuinely harder because action-angle coordinates are singular at the origin, so the conjugation schemes producing the torus examples have no direct analogue at a fixed point; the extension of analytic instability constructions from tori to elliptic equilibria is stated as an open and challenging problem in both sources. Before Fayad's work not a single real-analytic example of an unstable non-resonant elliptic equilibrium was known, and Arnold's classical expectation — recounted in Fayad's paper — that elliptic equilibria of generic real-analytic Hamiltonians should be Lyapunov unstable outside a few exceptional cases remains untested in low dimensions.

Problem Statement

Determine in which regimes real-analytic (in particular real entire) Hamiltonian systems can have a Lyapunov-unstable elliptic equilibrium — the extension of the analytic instability constructions known for quasi-periodic tori to elliptic equilibria, where action-angle coordinates are singular. The regimes left open by the known examples are:

  1. Convergent Birkhoff normal form (any number of degrees of freedom): does there exist a real-analytic Hamiltonian with a Lyapunov-unstable elliptic equilibrium whose BNF at the equilibrium converges? The quasi-periodic-torus analogue was answered positively by Farré-Fayad; the primary sources state the equilibrium case to be completely open in the real-analytic setting, and all known analytic examples have divergent BNF.

  2. Low dimensions and frequency arithmetic: does there exist a real-analytic Hamiltonian with two degrees of freedom (necessarily with a mixed-sign frequency vector, by the Lagrange-Dirichlet obstruction) having a Lyapunov-unstable elliptic equilibrium? Does one exist in three degrees of freedom with Diophantine frequency?

For each regime the question is settled either by an explicit construction — a concrete real-analytic Hamiltonian together with proofs that the origin is an elliptic equilibrium of the stated rationally independent frequency, that it is Lyapunov unstable (orbits starting arbitrarily near the origin eventually leave some fixed neighborhood), and that the BNF converges or diverges as claimed — or by a rigorous impossibility result establishing Lyapunov stability for every system in that regime, with the regularity, number of degrees of freedom, frequency arithmetic, and any hypothesis on the BNF exactly as in that regime, thereby showing that the regime admits no example.

References

  1. Gerard Farré and Bassam Fayad, Instabilities of invariant quasi-periodic tori, Journal of the European Mathematical Society 24 (2022) 4363-4383, DOI 10.4171/jems/1206, https://doi.org/10.4171/jems/1206 (preprint arXiv:1912.01575, https://arxiv.org/abs/1912.01575, under the title 'Instabilities for analytic quasi-periodic invariant tori')
  2. Bassam Fayad, Lyapunov unstable elliptic equilibria, Journal of the American Mathematical Society 36 (2023) 81-106, DOI 10.1090/jams/997, https://doi.org/10.1090/jams/997 (preprint arXiv:1809.09059, https://arxiv.org/abs/1809.09059)
  3. L. H. Eliasson, B. Fayad and R. Krikorian, Around the stability of KAM tori, Duke Mathematical Journal 164 (2015) 1733-1775, DOI 10.1215/00127094-3120060, https://doi.org/10.1215/00127094-3120060
  4. Frank Trujillo, Lyapunov instability in KAM stable Hamiltonians with two degrees of freedom, Journal of Modern Dynamics 19 (2023) 363-383, DOI 10.3934/jmd.2023010, https://doi.org/10.3934/jmd.2023010 (preprint arXiv:1912.12851, https://arxiv.org/abs/1912.12851)
  5. Pierre Berger, Analytic pseudo-rotations, Annals of Mathematics 204 (2026), no. 1, DOI 10.4007/annals.2026.204.1.3, https://doi.org/10.4007/annals.2026.204.1.3 (preprint arXiv:2210.03438, https://arxiv.org/abs/2210.03438)
  6. Pierre Berger, Analytic pseudo-rotations II: a principle for spheres, disks and annuli, preprint (2024), arXiv:2402.15303, https://arxiv.org/abs/2402.15303
  7. Abed Bounemoura and Gerard Farré, Positive measure of effective quasi-periodic motion near a Diophantine torus, Annales Henri Poincaré 24 (2023) 3289-3304, DOI 10.1007/s00023-023-01302-4, https://doi.org/10.1007/s00023-023-01302-4 (preprint arXiv:2105.01297, https://arxiv.org/abs/2105.01297)
  8. Bassam Fayad and Raphaël Krikorian, Some questions around quasi-periodic dynamics, Proceedings of the International Congress of Mathematicians (ICM 2018), Vol. III, 1909-1932, DOI 10.1142/9789813272880_0123, https://doi.org/10.1142/9789813272880_0123 (preprint arXiv:1809.10375, https://arxiv.org/abs/1809.10375)
  9. Santiago Barbieri and Gerard Farré, Nearly-optimal effective stability estimates around Diophantine tori of Hölder Hamiltonians, preprint (2024), arXiv:2402.10764, https://arxiv.org/abs/2402.10764