For a real-analytic Hamiltonian with a Diophantine elliptic equilibrium point, Bounemoura, Fayad and Niederman (Adv. Math. 2020) proved that, generically in the finite-order Taylor jet, orbits stay close to the equilibrium for a doubly exponentially long time T(r)≥exp⁡(exp⁡(Cr−1/(τ+1)))T(r)\ge\exp(\exp(Cr^{-1/(\tau+1)})), where rr is the initial distance to the equilibrium. Their paper poses three open questions that together ask for the sharp asymptotics and the full geometric scope of this stability theorem: (1) what is the optimal lower bound on T(r)T(r) when no hypothesis is made on the Birkhoff normal form, and do near-optimal analytic instability examples exist at Diophantine equilibria as they do for invariant tori; (2) is the double-exponential estimate essentially optimal in the analytic category (optimality is known only in Gevrey classes, for tori); (3) does the generic double-exponential stability extend to quasi-periodic normally elliptic reducible Diophantine invariant tori, beyond the Lagrangian case already settled by the same authors. All three cores remain open as of 2026.

Background

An equilibrium point of a Hamiltonian flow is called elliptic if the eigenvalues of the linearized vector field are all purely imaginary and simple; writing the quadratic part of the Hamiltonian as α⋅I(x,y)\alpha\cdot I(x,y), where I(x,y)=(x12+y12,…,xn2+yn2)/2I(x,y)=(x_1^2+y_1^2,\dots,x_n^2+y_n^2)/2 are the actions and α∈Rn\alpha\in\mathbb{R}^n is the frequency vector, ellipticity means the linearized motion is a superposition of oscillations with frequencies αj\alpha_j. The frequency vector is non-resonant if k⋅α≠0k\cdot\alpha\ne0 for all integer vectors k≠0k\ne0, and Diophantine (written α∈DC(τ,γ)\alpha\in DC(\tau,\gamma)) if ∣k⋅α∣≥γ∣k∣−τ|k\cdot\alpha|\ge\gamma|k|^{-\tau} for all k∈Zn∖{0}k\in\mathbb{Z}^n\setminus\{0\}; the Diophantine exponent τ≥n−1\tau\ge n-1 controls the severity of the small divisors k⋅αk\cdot\alpha that appear in every perturbative construction. By a classical theorem of Birkhoff, a non-resonant analytic Hamiltonian can be formally conjugated to a function of the actions alone, h(I)h(I); truncating at order KK gives the Birkhoff normal form of order KK (BNF), a uniquely defined polynomial in II whose remainder is of higher order. The formal series generically diverges, so one cannot in general conclude that the equilibrium is a center. Effective (quantitative) stability sidesteps this: define T(r)T(r) as the largest time such that every orbit starting within distance rr of the equilibrium remains within distance 2r2r for all ∣t∣≤T(r)|t|\le T(r). Lyapunov stability would correspond to T(r)=∞T(r)=\infty; Arnold conjectured that for n≥3n\ge3 degrees of freedom and indefinite quadratic part a generic analytic elliptic equilibrium is Lyapunov unstable, but only particular unstable examples are known (Fayad 2023, with divergent BNF). Quantitatively, classical averaging estimates (as recalled in the source paper) give, for every analytic Hamiltonian with α∈DC(τ,γ)\alpha\in DC(\tau,\gamma) and without any hypothesis on the normal form, a lower bound T(r)≳exp⁡((γr−1)1/(τ+1))T(r)\gtrsim\exp((\gamma r^{-1})^{1/(\tau+1)}) (singly exponential). Nekhoroshev theory upgrades such bounds under a geometric condition on the integrable part (steepness, a transversality condition on the gradient preventing overly fast drift of the actions; stably steep means the condition holds on a whole neighborhood in the space of polynomials). Combining a genericity result for stably steep finite-order BNFs with a Nekhoroshev-type estimate in a singularly perturbed regime (Giorgilli-Morbidelli mechanism for KAM (Kolmogorov-Arnold-Moser) tori), Bounemoura-Fayad-Niederman proved: generically, in the sense of an open dense set of full Lebesgue measure in the space of Taylor jets of order K0=n2+4K_0=n^2+4 (with the stably steep condition imposed on the BNF truncated at order m0=⌊(n2+4)/2⌋m_0=\lfloor(n^2+4)/2\rfloor), an elliptic equilibrium with α∈DC(τ,γ)\alpha\in DC(\tau,\gamma) satisfies T(r)≥exp⁡(exp⁡(Cr−1/(τ+1)))T(r)\ge\exp(\exp(Cr^{-1/(\tau+1)})) — double-exponential stability, the effective analogue of super-exponential stability known for invariant tori. The companion paper of the same authors proved the analogous statement for invariant Lagrangian (maximal-dimensional) Diophantine tori. A quasi-periodic normally elliptic reducible invariant torus interpolates between these two settings: in coordinates (θ,J,x,y)∈Tm×Rm×R2n(\theta,J,x,y)\in\mathbb{T}^m\times\mathbb{R}^m\times\mathbb{R}^{2n} the Hamiltonian reads H=β⋅J+α⋅I(x,y)+F(θ,I,x,y)H=\beta\cdot J+\alpha\cdot I(x,y)+F(\theta,I,x,y) with FF of order at least 2 in II and 3 in (x,y)(x,y); the set {J=0, I=0}\{J=0,\,I=0\} is a torus carrying quasi-periodic motion of internal frequency β\beta, the transverse dynamics is elliptic with frequency α\alpha, and reducible means the linearized transverse system reduces to constant coefficients. Whether generic double-exponential stability holds for the joint actions (J,I)(J,I) when (β,α)(\beta,\alpha) is Diophantine is the third question; Gevrey classes (regularity between C∞C^\infty and analytic, with derivative growth like k!σk!^{\sigma}) enter as the regularity in which matching unstable examples are currently constructible.

Problem Statement

Let HH be a real-analytic Hamiltonian on R2n\mathbb{R}^{2n} with an elliptic equilibrium point at the origin of non-resonant frequency vector α\alpha, and let T(r)T(r) be the effective stability time: the largest time such that every orbit starting within distance rr of the origin remains within distance 2r2r for all ∣t∣≤T(r)|t|\le T(r). Theorem A of Bounemoura-Fayad-Niederman (2020) states, writing K0=K0(n):=n2+4K_0=K_0(n):=n^2+4 and m0=⌊K0/2⌋m_0=\lfloor K_0/2\rfloor for the truncation orders, that if the Birkhoff normal form of order K0K_0 truncated at order m0m_0 (the polynomial part of degree at most m0m_0 of the formal BNF of HH) is stably steep (steepness holding on a whole neighborhood in the space of polynomials, as defined in the background) — a generic condition (the set of such jets is open, dense, and of full Lebesgue measure in the space P3(2n,K0)P_3(2n,K_0) of Taylor jets with terms of degrees 33 to K0K_0) — then for α∈DC(τ,γ)\alpha\in DC(\tau,\gamma) one has T(r)≥exp⁡(exp⁡(Cr−1/(τ+1)))T(r)\ge\exp(\exp(Cr^{-1/(\tau+1)})) for small rr. The open problem is the collection of the three questions the authors pose about this theorem, which share one scientific objective — the sharp asymptotics and the full geometric scope of generic effective stability for Diophantine elliptic fixed objects of analytic Hamiltonians:

(Q1) What are the optimal lower bounds on T(r)T(r) if no assumption whatsoever is made on the Birkhoff normal form, beyond α∈DC(τ,γ)\alpha\in DC(\tau,\gamma)? The only unconditional bound is the classical singly-exponential T(r)≳exp⁡((γr−1)1/(τ+1))T(r)\gtrsim\exp((\gamma r^{-1})^{1/(\tau+1)}); in particular, do there exist real-analytic Hamiltonians with a Diophantine elliptic equilibrium for which the escape time satisfies an upper bound of essentially the same shape (matching the exponent 1/(τ+1)1/(\tau+1) up to an arbitrarily small ε>0\varepsilon>0), as is known for invariant tori by Farré-Fayad? No such example is currently known for equilibria.

(Q2) Is the double-exponential lower bound of Theorem A essentially optimal in the analytic category — i.e. can one construct analytic Hamiltonians satisfying the generic hypothesis (or show the estimate cannot be improved) for which escape from a distance-rr neighborhood occurs within a time not exceeding a double exponential of order r−1/(τ+1)r^{-1/(\tau+1)}? Optimality is expected and known only in weaker regularity (Gevrey examples of Fayad-Sauzin for quasi-convex tori) and in quasi-convex settings; the analytic case is unresolved.

(Q3) For a quasi-periodic normally elliptic reducible invariant torus, described in action-angle and transverse symplectic coordinates (θ,J,x,y)∈Tm×Rm×R2n(\theta,J,x,y)\in\mathbb{T}^m\times\mathbb{R}^m\times\mathbb{R}^{2n} by

H(θ,J,x,y)=β⋅J+α⋅I(x,y)+F(θ,I,x,y),H(\theta,J,x,y)=\beta\cdot J+\alpha\cdot I(x,y)+F(\theta,I,x,y),

with FF real-analytic and of order at least 2 in II and 3 in (x,y)(x,y), and with the joint vector (β,α)∈Rm+n(\beta,\alpha)\in\mathbb{R}^{m+n} Diophantine of exponent τ\tau: does the conclusion of Theorem A hold for generic FF — i.e. if (J(0),I(x(0),y(0)))(J(0),I(x(0),y(0))) is at distance rr of the origin in Rn+m\mathbb{R}^{n+m}, with rr sufficiently small, does (J(t),I(x(t),y(t)))(J(t),I(x(t),y(t))) remain within distance 2r2r of the origin for a time T(r)T(r) that is doubly exponentially large with respect to r−1/(τ+1)r^{-1/(\tau+1)}? The Lagrangian case (n=0n=0-transverse, pure action variables) was resolved positively by the same authors in a companion paper; the general normally elliptic case, which contains both elliptic equilibria and Lagrangian tori as limiting cases, is open.

Q1, Q2 and Q3 are independent — each is a self-contained question that can be settled on its own, and settling one of them leaves the other two open — but the objective above is their union, including both parts of Q1 (the optimal unconditional lower bound on T(r)T(r) and the existence of examples matching that bound).

References

  1. Abed Bounemoura, Bassam Fayad, Laurent Niederman, "Super-exponential stability for generic real-analytic elliptic equilibrium points", Advances in Mathematics 366 (2020), 107088. DOI: 10.1016/j.aim.2020.107088, https://doi.org/10.1016/j.aim.2020.107088 (source paper)
  2. Abed Bounemoura, Bassam Fayad, Laurent Niederman, "Superexponential Stability of Quasi-Periodic Motion in Hamiltonian Systems", Communications in Mathematical Physics 350 (2017), 361-386. DOI: 10.1007/s00220-016-2782-9, https://doi.org/10.1007/s00220-016-2782-9 (companion paper resolving the Lagrangian-torus case of Question 3)
  3. Abed Bounemoura, Bassam Fayad, Laurent Niederman, "Nekhoroshev estimates for steep real-analytic elliptic equilibrium points", Nonlinearity 33 (2020), 1-33. DOI: 10.1088/1361-6544/ab4c89, https://doi.org/10.1088/1361-6544/ab4c89 (companion paper proving the Nekhoroshev-type input for steep equilibria)
  4. Alessandro Morbidelli, Antonio Giorgilli, "Superexponential stability of KAM tori", Journal of Statistical Physics 78 (1995), 1607-1617. DOI: 10.1007/BF02180145, https://doi.org/10.1007/BF02180145 (origin of the combined BNF/Nekhoroshev mechanism)
  5. Laurent Niederman, "Generic super-exponential stability of elliptic equilibrium positions for symplectic vector fields", Regular and Chaotic Dynamics 18 (2013), 719-731. DOI: 10.1134/S1560354713060117, https://doi.org/10.1134/S1560354713060117 (prior super-exponential result under non-generic Kolmogorov non-degeneracy)
  6. 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 (also arXiv:1809.09059; particular analytic unstable examples with divergent BNF)
  7. Gerard Farré, 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 (also arXiv:1912.01575; analytic unstable tori of arbitrary frequency and optimality of exponential stability for Diophantine tori)
  8. Bassam Fayad, David Sauzin, "KAM Tori are No More than Sticky", Archive for Rational Mechanics and Analysis 237 (2020), 1177-1211. DOI: 10.1007/s00205-020-01526-2, https://doi.org/10.1007/s00205-020-01526-2 (also arXiv:1812.04163; Gevrey examples showing optimality of double-exponential stability for quasi-convex tori)
  9. Gerard Farré, "On the optimal effective stability bounds for quasi-periodic tori of finitely differentiable and Gevrey Hamiltonians", Archiv der Mathematik 122 (2023), 213-225. DOI: 10.1007/s00013-023-01928-0, https://doi.org/10.1007/s00013-023-01928-0 (near-optimality of effective stability exponents for Gevrey and finitely differentiable tori)
  10. Abed Bounemoura, 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 (double-exponentially long effective quasi-periodic motion on a positive-measure set near a Diophantine torus, weakened Herman conjecture)
  11. Massimiliano Guzzo, Chiara Caracciolo, Gabriella Pinzari, "Improved stability estimates at elliptic equilibria of Hamiltonian systems", Discrete and Continuous Dynamical Systems 52 (2026), 575-605. DOI: 10.3934/dcds.2026063, https://doi.org/10.3934/dcds.2026063 (also arXiv:2601.18268; improved a-priori lower bounds from a finite-order BNF)
  12. 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 (also arXiv:1912.12851; Gevrey KAM-stable yet Lyapunov-unstable elliptic equilibria in two degrees of freedom)
  13. Raphaël Krikorian, "On the divergence of Birkhoff Normal Forms", Publications Mathématiques de l'IHÉS 135 (2022), 1-181. DOI: 10.1007/s10240-022-00130-2, https://doi.org/10.1007/s10240-022-00130-2 (genericity of BNF divergence in related quasi-periodic settings)
  14. Chiara Caracciolo, "Normal form for lower dimensional elliptic tori in Hamiltonian systems", arXiv:2110.09824 (2021), https://arxiv.org/abs/2110.09824 (convergent KAM normal-form construction for lower-dimensional elliptic tori; persistence, not stability times)
  15. Donato Scarcella, Frank Trujillo, "Non-autonomous KAM theory for lower dimensional invariant tori (I): Normally elliptic and hyperbolic cases", arXiv:2607.28452 (2026), https://arxiv.org/abs/2607.28452 (current activity on normally elliptic tori under decaying non-autonomous perturbations; not effective stability times)