Effective stability estimates close to resonances with applications to rotational dynamics

Alessandra Celletti; Alessia Francesca Guido; Anargyros Dogkas

arxiv: 2604.04955 · v1 · submitted 2026-04-03 · 🧮 math.DS · astro-ph.EP· math-ph· math.MP

Effective stability estimates close to resonances with applications to rotational dynamics

Alessandra Celletti , Anargyros Dogkas , Alessia Francesca Guido This is my paper

Pith reviewed 2026-05-13 18:30 UTC · model grok-4.3

classification 🧮 math.DS astro-ph.EPmath-phmath.MP MSC 37J40

keywords Nekhoroshev estimatesresonancesspin-orbit problemstability estimatesHamiltonian systemscelestial mechanicsperturbation theoryDiophantine frequencies

0 comments

The pith

Nekhoroshev-like estimates with optimized parameters and reduced perturbations provide effective stability for orbits near resonances in spin-orbit models.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to obtain effective stability bounds for action variables in nearly-integrable Hamiltonian systems near resonances by applying Nekhoroshev-like estimates. Parameters in these estimates are chosen through an optimization algorithm to maximize the stability time. Perturbation theory is used to further decrease the size of the perturbing function. This approach is applied to the spin-orbit problem and the spin-spin-orbit model, demonstrating stability for orbits with frequencies approaching resonant values. Such results matter because they offer concrete time scales over which rotational motions in celestial systems remain predictable and bounded near resonances.

Core claim

In nearly-integrable Hamiltonian systems defined over non-resonant domains, Nekhoroshev-like estimates are applied in the neighborhood of resonances to bound the variation of the action variables over long times. An optimization algorithm selects parameters to maximize the stability time, and perturbation theory reduces the norm of the perturbing function. When implemented for the one-dimensional spin-orbit problem and the two-dimensional spin-spin-orbit model, this yields stability results for sequences of Diophantine frequencies converging to the main resonances of these rotational dynamics models.

What carries the argument

Nekhoroshev-like estimates applied near resonances, with parameter optimization and perturbation theory to minimize the perturbing function norm.

If this is right

Explicit long-term bounds on action drift are obtained for the spin-orbit problem near its main resonances.
The two-dimensional spin-spin-orbit model admits similar stability estimates close to its resonances.
Stability holds for Diophantine frequencies arbitrarily close to resonant ones after optimization.
The freedom in parameter choice allows tailoring the estimates to specific models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The technique could be extended to other nearly-integrable systems in celestial mechanics, such as those with more degrees of freedom.
Further reductions in perturbation norms might be achieved by higher-order perturbation theory, leading to even longer stability times.
These bounds suggest that numerical explorations of rotational dynamics should focus on verifying stability within the predicted intervals.
Applications might include assessing the long-term behavior of planetary spins in multi-body systems.

Load-bearing premise

The systems are nearly-integrable and the Nekhoroshev-like estimates apply in the resonance neighborhoods with parameters that can be freely chosen and optimized.

What would settle it

Direct numerical simulation of the spin-orbit Hamiltonian showing that the action variables change by more than the estimated bound within the computed stability time for a frequency close to resonance would falsify the result.

Figures

Figures reproduced from arXiv: 2604.04955 by Alessandra Celletti, Alessia Francesca Guido, Anargyros Dogkas.

**Figure 1.** Figure 1: Stability time for z = 2, . . . , 100 with s = 1.6 (1 : 1 resonance) and s = 0.6 (3 : 2 resonance) for (a) ε = 10−4 after two perturbative steps, (b) ε = 10−3 after two perturbative steps, (c) ε = 10−3 after three perturbative steps, (d) ε = 10−2 after three perturbative steps (d). where stability results are affected, even if there are no main or secondary resonances nearby. These regions are an indirect … view at source ↗

**Figure 2.** Figure 2: Comparison between the last value of p1 for which the algorithm provides results (blue line) and a resonant normal form approximation of the width of the 1:1 resonance (red line) for (a) ε = 10−4 after two perturbative steps, (b) ε = 10−3 after two perturbative steps, (c) ε = 10−3 after three perturbative steps, (d) ε = 10−2 after three perturbative steps. radial grid of initial conditions (p1(0), p2(0))z,… view at source ↗

**Figure 3.** Figure 3: Comparison between the last value of p1 for which the algorithm provides results (blue line) and a resonant normal form approximation of the width of the 3:2 resonance (red line) for (a) ε = 10−4 after two perturbative step, (b) ε = 10−3 after two perturbative steps, (c) ε = 10−3 after three perturbative steps, (d) ε = 10−2 after three perturbative steps. computed by implementing the algorithm. The red poi… view at source ↗

**Figure 4.** Figure 4: Stability time around the spin-spin-orbit resonance (1 : 1)S1 ,(3 : 2)S2 , for ε1 = ε2 = 10−5 after: (a) one perturbative step, (b) two perturbative steps. Stability time around the spin-spin-orbit resonance: (c) (1 : 1)S1 ,(3 : 2)S2 , for ε1 = 3 · 10−5 and ε2 = 10−4 after two perturbative steps, (d) (1 : 1)S1 ,(1 : 1)S2 , for ε1 = ε2 = 3 · 10−5 after two perturbative steps. condition, failure points may n… view at source ↗

**Figure 5.** Figure 5: Bound on the actions around the spin-spin-orbit resonance (1 : 1)S1 ,(3 : 2)S2 , for ε1 = ε2 = 10−5 after: (a) one perturbative step, (b) two perturbative steps. Bound on the actions around the spin-spin-orbit resonance: (c) (1 : 1)S1 ,(3 : 2)S2 , for ε1 = 3 · 10−5 and ε2 = 10−4 after two perturbative steps, (d) (1 : 1)S1 ,(1 : 1)S2 , for ε1 = ε2 = 3 · 10−5 after two perturbative steps. of them. The points… view at source ↗

**Figure 6.** Figure 6: The values of T (left), K (middle), and dK (right) for every step of the optimization process. We indicate with green points the optimization steps that found a new largest value of T, with blue the accepted in-between steps, which satisfy the conditions but do not find a new largest T, and with red the rejected steps, which do not satisfy at least one of the conditions of Theorem 1. We display two initia… view at source ↗

read the original abstract

We consider nearly-integrable Hamiltonian systems defined over a non-resonant domain. In the neighborhood of resonances, we use Nekhoroshev-like estimates to provide effective stability bounds for the action variables over long time. The applicability conditions of these estimates allow some freedom in the choice of parameters. Hence, we develop an optimization algorithm for choosing parameters that maximize the stability time. To further improve the stability estimates, we use perturbation theory to reduce the norm of the perturbing function. We implement this procedure (effective stability estimates and perturbation theory) to analyze the stability of sequences of irrational (Diophantine) frequencies converging to frequencies corresponding to resonances. We consider two applications to models describing problems of rotational dynamics in Celestial Mechanics: the spin-orbit problem, described by a 1D time-dependent Hamiltonian, and the spin-spin-orbit model, described by a 2D time-dependent Hamiltonian. We show stability results for orbits close to the main resonances associated with such models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds a parameter-optimization step to Nekhoroshev estimates near resonances and applies it to spin-orbit and spin-spin-orbit models, but the validity of the optimized bounds needs explicit checking.

read the letter

The new piece is the algorithm that tunes the Nekhoroshev parameters (epsilon, tau, and the like) to push the stability time as high as possible once a preliminary perturbation reduction has been done. They then run this on sequences of Diophantine frequencies approaching the main resonances in the 1D spin-orbit problem and the 2D spin-spin-orbit problem, producing concrete lower bounds on how long the actions stay close to their initial values. That is a legitimate, if incremental, extension of existing Nekhoroshev work to these rotational models, and the concrete numbers will be useful to people who actually integrate or analyze these Hamiltonians in celestial mechanics. The perturbation-reduction step is standard and helps shrink the effective norm before the Nekhoroshev theorem is applied, which is the right direction. The optimization itself is described at a high level in the abstract, so the main question is whether the chosen parameter values are checked afterward to confirm they still lie inside the domain where the Nekhoroshev hypotheses hold. If the optimization is allowed to increase the effective perturbation size beyond the threshold set by the distance to resonance and the Diophantine constants, the stability times lose their justification. The abstract does not say whether such a verification is performed or reported. On the evidence available, the central claim is not circular, but the gap between the algorithmic freedom and the theorem's smallness requirement is the soft spot that needs to be closed in the full text. This is a solid subfield paper that supplies usable bounds for rotational dynamics; it is not reshaping the broader theory. I would bring it to a reading group focused on celestial mechanics or effective stability, would not cite it in my own work outside that niche, and would send it to referees rather than desk-reject it.

Referee Report

3 major / 3 minor

Summary. The manuscript develops effective stability estimates for action variables in nearly-integrable Hamiltonian systems near resonances by applying Nekhoroshev-like theorems, optimizing free parameters (such as ε and τ) to maximize stability times, and using perturbation theory to reduce the perturbing norm. The procedure is implemented for sequences of Diophantine frequencies approaching resonant ones and applied to the 1D spin-orbit and 2D spin-spin-orbit models, yielding stability results for orbits close to the main resonances in these celestial mechanics problems.

Significance. If the post-optimization validity conditions hold, the work supplies concrete, computable stability times for rotational dynamics near resonances, which is of practical value in celestial mechanics. The combination of parameter optimization with a preliminary perturbation reduction is a methodological strength that can improve upon direct Nekhoroshev application; the explicit algorithmic treatment and concrete model applications add to the utility.

major comments (3)

[§4] §4 (Optimization procedure): The optimization algorithm for Nekhoroshev parameters is described at a high level and stated to maximize stability time, but no a-posteriori verification is supplied that the selected values keep the reduced perturbation norm below the smallness threshold required by the cited Nekhoroshev-like estimates (relative to resonance distance and Diophantine constants). This is load-bearing for the central claim.
[§5] §5 (Spin-orbit application): The stability bounds for Diophantine frequency sequences converging to the main resonances are presented after optimization, yet the manuscript does not report the concrete optimized parameter values nor confirm they remain inside the domain of validity of the underlying theorem; without this, the reported stability times cannot be directly justified.
[§3.1–3.2] §3.1–3.2 (Nekhoroshev estimates near resonances): The claim that applicability conditions 'allow some freedom' in parameter choice is used to justify optimization, but the text does not quantify how the optimization step alters the effective perturbation size or provide an explicit check against the theorem hypotheses after reduction; this leaves a moderate derivation gap between the abstract statement and the final bounds.

minor comments (3)

[Notation] The notation for Diophantine constants, effective perturbation norms, and stability times is introduced piecemeal; a consolidated table of symbols with references to their first appearance would improve readability.
[Figures] Figure captions for the frequency sequences and stability plots lack explicit indication of the optimized parameter values used to generate each curve.
[§2] A few references to standard Nekhoroshev statements (e.g., the precise form of the stability time exponent) could be added for completeness in §2.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough reading and valuable comments, which help clarify the presentation of our optimization procedure and its verification. We address each major comment below and will revise the manuscript to incorporate the requested details and checks.

read point-by-point responses

Referee: [§4] §4 (Optimization procedure): The optimization algorithm for Nekhoroshev parameters is described at a high level and stated to maximize stability time, but no a-posteriori verification is supplied that the selected values keep the reduced perturbation norm below the smallness threshold required by the cited Nekhoroshev-like estimates (relative to resonance distance and Diophantine constants). This is load-bearing for the central claim.

Authors: We agree that an explicit a-posteriori verification is essential for the central claims. In the revised manuscript we will add a dedicated subsection (or appendix) that computes the reduced perturbation norm after optimization for each frequency sequence and directly compares it to the smallness threshold of the Nekhoroshev-like theorem, taking into account the resonance distance and the specific Diophantine constants. This verification will be performed numerically for the sequences considered in the applications. revision: yes
Referee: [§5] §5 (Spin-orbit application): The stability bounds for Diophantine frequency sequences converging to the main resonances are presented after optimization, yet the manuscript does not report the concrete optimized parameter values nor confirm they remain inside the domain of validity of the underlying theorem; without this, the reported stability times cannot be directly justified.

Authors: We will include a table (or set of tables) in the revised §5 that lists the concrete optimized values of the free parameters (ε, τ and any auxiliary constants) for each Diophantine frequency in the sequences approaching the main resonances. For each entry we will also report the corresponding reduced perturbation norm and explicitly confirm that it lies below the theorem threshold, thereby justifying the reported stability times. revision: yes
Referee: [§3.1–3.2] §3.1–3.2 (Nekhoroshev estimates near resonances): The claim that applicability conditions 'allow some freedom' in parameter choice is used to justify optimization, but the text does not quantify how the optimization step alters the effective perturbation size or provide an explicit check against the theorem hypotheses after reduction; this leaves a moderate derivation gap between the abstract statement and the final bounds.

Authors: We will expand §§3.1–3.2 to quantify the effect of optimization on the effective perturbation size (e.g., by tabulating or plotting the norm before and after the optimization step) and to supply explicit post-reduction checks against all hypotheses of the Nekhoroshev-like estimates. These additions will close the derivation gap and make the transition from the abstract theorem to the concrete bounds fully transparent. revision: yes

Circularity Check

0 steps flagged

Stability estimates derived from external Nekhoroshev theorems via algorithmic parameter optimization

full rationale

The paper applies Nekhoroshev-like estimates to resonant neighborhoods after an algorithmic optimization of free parameters (chosen to maximize the resulting stability time) and a preliminary perturbation-theory reduction of the perturbing norm. The stability bounds are direct consequences of the cited external theorems evaluated at the optimized values, without any redefinition of the target stability time in terms of itself, without fitting parameters to the final result, and without load-bearing self-citations that reduce the central claim to an unverified premise. The derivation chain remains self-contained against the stated applicability conditions of the theorems.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The estimates rest on the standard hypotheses of Nekhoroshev theory plus the existence of an optimization routine that selects parameters without introducing new fitted constants beyond those already in the theorem.

free parameters (1)

Nekhoroshev parameters (epsilon, tau, etc.)
Free parameters inside the Nekhoroshev-type bounds that are varied by the optimization algorithm to maximize the stability time.

axioms (1)

domain assumption The Hamiltonian is nearly integrable and the frequency vector satisfies the non-resonance conditions outside a small neighborhood of the target resonances.
Invoked to justify application of Nekhoroshev-like estimates in the neighborhood of resonances.

pith-pipeline@v0.9.0 · 5474 in / 1304 out tokens · 37202 ms · 2026-05-13T18:30:25.410486+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

we develop an optimization algorithm for choosing parameters that maximize the stability time... sequences of irrational (Diophantine) frequencies... Γ(k1,k2)z,s = −k2/k1 − s/(z+γ) with γ golden ratio
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1... E ≤ 1/(27ℓ) α r / K ... T ≡ s0 r / (5 E) exp(K s0 / 6)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

[1]

V. I. Arnol’d. Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations. Russian Math. Surveys, 18(5):9–36, 1963

work page 1963
[2]

G. Bou´ e. The two rigid body interaction using angular momentum theory formulae. Celestial Mechanics and Dynamical Astronomy, 128(2-3):261–273, June 2017

work page 2017
[3]

Bounemoura and J.-P

A. Bounemoura and J.-P. Marco. Improved exponential stability for near-integrable quasi-convex hamiltonians. Nonlinearity, 24(1):97, nov 2010

work page 2010
[4]

A. P. Bustamante, A. Celletti, and C. Lhotka. The dynamics of the spin–spin problem in Celestial Mechanics. Communications in Nonlinear Science and Numerical Simulations, 142:108548, Mar. 2025

work page 2025
[5]

J. Cassels. An introduction to Diophantine approximation. Cambridge University Press, 1957

work page 1957
[6]

Celletti

A. Celletti. Analysis of resonances in the spin-orbit problem in celestial mechanics: the synchronous resonance. I. Z. Angew. Math. Phys., 41(2):174–204, 1990

work page 1990
[7]

Celletti

A. Celletti. Stability and Chaos in Celestial Mechanics. Springer-Verlag, Berlin; published in asso- ciation with Praxis Publishing, Chichester, 2010

work page 2010
[8]

Celletti, I

A. Celletti, I. De Blasi, and C. Efthymiopoulos. Nekhoroshev estimates for the orbital stability of Earth’s satellites. Celestial Mechanics and Dynamical Astronomy, 135(2):10, Apr. 2023

work page 2023
[9]

Celletti, C

A. Celletti, C. Falcolini, and U. Locatelli. On the break-down threshold of invariant tori in four dimensional maps. Regul. Chaotic Dyn., 9(3):227–253, 2004

work page 2004
[10]

Celletti and L

A. Celletti and L. Ferrara. An Application of the Nekhoroshev Theorem to the Restricted Three- Body Problem. Celestial Mechanics and Dynamical Astronomy, 64(3):261–272, Sept. 1996

work page 1996
[11]

Celletti, J

A. Celletti, J. Gimeno, and M. Misquero. The Spin-Spin Problem in Celestial Mechanics. Journal of NonLinear Science, 32(6):88, Dec. 2022

work page 2022
[12]

Celletti and A

A. Celletti and A. Giorgilli. On the stability of the Lagrangian points in the spatial restricted problem of three bodies. Celestial Mechanics and Dynamical Astronomy, 50(1):31–58, Mar. 1990

work page 1990
[13]

Chirikov

B. Chirikov. A universal instability of many-dimensional oscillator systems. Phys. Rep., 52(5):264– 379, 1979

work page 1979
[14]

Efthymiopoulos

C. Efthymiopoulos. Canonical perturbation theory; stability and diffusion in Hamiltonian systems: applications in dynamical astronomy. Workshop Series of the Asociacion Argentina de Astronomia, 3:3–146, Jan. 2011. 38 A. CELLETTI, A. DOGKAS, AND A. FRANCESCA GUIDO

work page 2011
[15]

Efthymiopoulos, A

C. Efthymiopoulos, A. Giorgilli, and G. Contopoulos. Nonconvergence of formal integrals: II. Im- proved estimates for the optimal order of truncation. Journal of Physics A Mathematical General, 37(45):10831–10858, Nov. 2004

work page 2004
[16]

Ferraz-Mello

S. Ferraz-Mello. Canonical Perturbation Theories - Degenerate Systems and Resonance, volume

work page
[17]

Giorgilli and C

A. Giorgilli and C. Skokos. On the stability of the Trojan asteroids. Astron. & Astroph., 317:254– 261, Jan. 1997

work page 1997
[18]

Goldreich and S

P. Goldreich and S. Peale. Spin-orbit coupling in the solar system. Astronomical Journal, Vol. 71, p. 425 (1966), 71:425, 1966

work page 1966
[19]

J. M. Greene. A method for determining a stochastic transition. Jour. Math. Phys., 20:1183–1201, 1979

work page 1979
[20]

Guzzo, L

M. Guzzo, L. Chierchia, and G. Benettin. The steep Nekhoroshev’s Theorem and optimal stability exponents (an announcement). Rendiconti Lincei. Scienze Fisiche e Naturali, 25(3):293–299, Aug. 2014

work page 2014
[21]

Hagihara

Y. Hagihara. Celestial mechanics. Vol.1: Dynamical principles and transformation theory. 1970

work page 1970
[22]

Khinchin

A. Khinchin. Continued Fractions. Dover Publications, Mineola, New York, 1964

work page 1964
[23]

A. N. Kolmogorov. On conservation of conditionally periodic motions for a small change in Hamil- ton’s function. Dokl. Akad. Nauk SSSR (N.S.), 98:527–530, 1954. English translation inStochastic Behavior in Classical and Quantum Hamiltonian Systems (Volta Memorial Conf., Como, 1977), Lecture Notes in Phys., 93, pages 51–56. Springer, Berlin, 1979

work page 1954
[24]

H. Lei. Spin–orbit coupling of the primary body in a binary asteroid system. Celestial Mechanics and Dynamical Astronomy, 136(5):37, Oct. 2024

work page 2024
[25]

P. Lochak. Canonical perturbation theory via simultaneous approximation. Russian Mathematical Surveys, 47(6):57, dec 1992

work page 1992
[26]

Lochak and A

P. Lochak and A. I. Neishtadt. Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian. Chaos, 2(4):495–499, Oct. 1992

work page 1992
[27]

A. J. Maciejewski. Reduction, Relative Equilibria and Potential in the Two Rigid Bodies Problem. Celestial Mechanics and Dynamical Astronomy, 63(1):1–28, Mar. 1995

work page 1995
[28]

Misquero

M. Misquero. The spin-spin model and the capture into the double synchronous resonance. Nonlinearity, 34(4):2191–2219, Apr. 2021

work page 2021
[29]

J. Moser. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. G¨ ottingenMath.-Phys. Kl. II, 1962:1–20, 1962

work page 1962
[30]

N. N. Nekhoroshev. An exponential estimate of the time of stability of nearly integrable Hamil- tonian systems. Uspehi Mat. Nauk, 32(6(198)):5–66, 287, 1977. English translation: Russian Math. Surveys, 32(6):1–65, 1977

work page 1977
[31]

Pinzari, B

G. Pinzari, B. Scoppola, and M. Veglianti. Spin orbit resonance cascade via core shell model: application to Mercury and Ganymede. Celestial Mechanics and Dynamical Astronomy, 136(5):39, Oct. 2024

work page 2024
[32]

P¨ oschel

J. P¨ oschel. Nekhoroshev estimates for quasi-convex Hamiltonian systems.Math. Z., 213(2):187–216, 1993

work page 1993
[33]

E. T. Whittaker and G. N. Watson. A Course of Modern Analysis. Cambridge University Press, 5 edition, 2021

work page 2021
[34]

Wisdom, S

J. Wisdom, S. J. Peale, and F. Mignard. The chaotic rotation of Hyperion. Icarus, 58(2):137–152, May 1984

work page 1984
[35]

Zhang and K

J. Zhang and K. Zhang. Improved stability for analytic quasi-convex nearly integrable systems and optimal speed of arnold diffusion. Nonlinearity, 30(7):2918, jun 2017

work page 2017
[36]

K. Zhang. Speed of arnold diffusion for analytic hamiltonian systems. Inventiones Mathematicae, 186:255–290, 11 2011. EFFECTIVE STABILITY ESTIMATES CLOSE TO RESONANCES 39 Department of Mathematics, University of Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma (Italy) Email address:celletti@mat.uniroma2.it Department of Mathematics, Universit...

work page 2011

[1] [1]

V. I. Arnol’d. Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations. Russian Math. Surveys, 18(5):9–36, 1963

work page 1963

[2] [2]

G. Bou´ e. The two rigid body interaction using angular momentum theory formulae. Celestial Mechanics and Dynamical Astronomy, 128(2-3):261–273, June 2017

work page 2017

[3] [3]

Bounemoura and J.-P

A. Bounemoura and J.-P. Marco. Improved exponential stability for near-integrable quasi-convex hamiltonians. Nonlinearity, 24(1):97, nov 2010

work page 2010

[4] [4]

A. P. Bustamante, A. Celletti, and C. Lhotka. The dynamics of the spin–spin problem in Celestial Mechanics. Communications in Nonlinear Science and Numerical Simulations, 142:108548, Mar. 2025

work page 2025

[5] [5]

J. Cassels. An introduction to Diophantine approximation. Cambridge University Press, 1957

work page 1957

[6] [6]

Celletti

A. Celletti. Analysis of resonances in the spin-orbit problem in celestial mechanics: the synchronous resonance. I. Z. Angew. Math. Phys., 41(2):174–204, 1990

work page 1990

[7] [7]

Celletti

A. Celletti. Stability and Chaos in Celestial Mechanics. Springer-Verlag, Berlin; published in asso- ciation with Praxis Publishing, Chichester, 2010

work page 2010

[8] [8]

Celletti, I

A. Celletti, I. De Blasi, and C. Efthymiopoulos. Nekhoroshev estimates for the orbital stability of Earth’s satellites. Celestial Mechanics and Dynamical Astronomy, 135(2):10, Apr. 2023

work page 2023

[9] [9]

Celletti, C

A. Celletti, C. Falcolini, and U. Locatelli. On the break-down threshold of invariant tori in four dimensional maps. Regul. Chaotic Dyn., 9(3):227–253, 2004

work page 2004

[10] [10]

Celletti and L

A. Celletti and L. Ferrara. An Application of the Nekhoroshev Theorem to the Restricted Three- Body Problem. Celestial Mechanics and Dynamical Astronomy, 64(3):261–272, Sept. 1996

work page 1996

[11] [11]

Celletti, J

A. Celletti, J. Gimeno, and M. Misquero. The Spin-Spin Problem in Celestial Mechanics. Journal of NonLinear Science, 32(6):88, Dec. 2022

work page 2022

[12] [12]

Celletti and A

A. Celletti and A. Giorgilli. On the stability of the Lagrangian points in the spatial restricted problem of three bodies. Celestial Mechanics and Dynamical Astronomy, 50(1):31–58, Mar. 1990

work page 1990

[13] [13]

Chirikov

B. Chirikov. A universal instability of many-dimensional oscillator systems. Phys. Rep., 52(5):264– 379, 1979

work page 1979

[14] [14]

Efthymiopoulos

C. Efthymiopoulos. Canonical perturbation theory; stability and diffusion in Hamiltonian systems: applications in dynamical astronomy. Workshop Series of the Asociacion Argentina de Astronomia, 3:3–146, Jan. 2011. 38 A. CELLETTI, A. DOGKAS, AND A. FRANCESCA GUIDO

work page 2011

[15] [15]

Efthymiopoulos, A

C. Efthymiopoulos, A. Giorgilli, and G. Contopoulos. Nonconvergence of formal integrals: II. Im- proved estimates for the optimal order of truncation. Journal of Physics A Mathematical General, 37(45):10831–10858, Nov. 2004

work page 2004

[16] [16]

Ferraz-Mello

S. Ferraz-Mello. Canonical Perturbation Theories - Degenerate Systems and Resonance, volume

work page

[17] [17]

Giorgilli and C

A. Giorgilli and C. Skokos. On the stability of the Trojan asteroids. Astron. & Astroph., 317:254– 261, Jan. 1997

work page 1997

[18] [18]

Goldreich and S

P. Goldreich and S. Peale. Spin-orbit coupling in the solar system. Astronomical Journal, Vol. 71, p. 425 (1966), 71:425, 1966

work page 1966

[19] [19]

J. M. Greene. A method for determining a stochastic transition. Jour. Math. Phys., 20:1183–1201, 1979

work page 1979

[20] [20]

Guzzo, L

M. Guzzo, L. Chierchia, and G. Benettin. The steep Nekhoroshev’s Theorem and optimal stability exponents (an announcement). Rendiconti Lincei. Scienze Fisiche e Naturali, 25(3):293–299, Aug. 2014

work page 2014

[21] [21]

Hagihara

Y. Hagihara. Celestial mechanics. Vol.1: Dynamical principles and transformation theory. 1970

work page 1970

[22] [22]

Khinchin

A. Khinchin. Continued Fractions. Dover Publications, Mineola, New York, 1964

work page 1964

[23] [23]

A. N. Kolmogorov. On conservation of conditionally periodic motions for a small change in Hamil- ton’s function. Dokl. Akad. Nauk SSSR (N.S.), 98:527–530, 1954. English translation inStochastic Behavior in Classical and Quantum Hamiltonian Systems (Volta Memorial Conf., Como, 1977), Lecture Notes in Phys., 93, pages 51–56. Springer, Berlin, 1979

work page 1954

[24] [24]

H. Lei. Spin–orbit coupling of the primary body in a binary asteroid system. Celestial Mechanics and Dynamical Astronomy, 136(5):37, Oct. 2024

work page 2024

[25] [25]

P. Lochak. Canonical perturbation theory via simultaneous approximation. Russian Mathematical Surveys, 47(6):57, dec 1992

work page 1992

[26] [26]

Lochak and A

P. Lochak and A. I. Neishtadt. Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian. Chaos, 2(4):495–499, Oct. 1992

work page 1992

[27] [27]

A. J. Maciejewski. Reduction, Relative Equilibria and Potential in the Two Rigid Bodies Problem. Celestial Mechanics and Dynamical Astronomy, 63(1):1–28, Mar. 1995

work page 1995

[28] [28]

Misquero

M. Misquero. The spin-spin model and the capture into the double synchronous resonance. Nonlinearity, 34(4):2191–2219, Apr. 2021

work page 2021

[29] [29]

J. Moser. On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. G¨ ottingenMath.-Phys. Kl. II, 1962:1–20, 1962

work page 1962

[30] [30]

N. N. Nekhoroshev. An exponential estimate of the time of stability of nearly integrable Hamil- tonian systems. Uspehi Mat. Nauk, 32(6(198)):5–66, 287, 1977. English translation: Russian Math. Surveys, 32(6):1–65, 1977

work page 1977

[31] [31]

Pinzari, B

G. Pinzari, B. Scoppola, and M. Veglianti. Spin orbit resonance cascade via core shell model: application to Mercury and Ganymede. Celestial Mechanics and Dynamical Astronomy, 136(5):39, Oct. 2024

work page 2024

[32] [32]

P¨ oschel

J. P¨ oschel. Nekhoroshev estimates for quasi-convex Hamiltonian systems.Math. Z., 213(2):187–216, 1993

work page 1993

[33] [33]

E. T. Whittaker and G. N. Watson. A Course of Modern Analysis. Cambridge University Press, 5 edition, 2021

work page 2021

[34] [34]

Wisdom, S

J. Wisdom, S. J. Peale, and F. Mignard. The chaotic rotation of Hyperion. Icarus, 58(2):137–152, May 1984

work page 1984

[35] [35]

Zhang and K

J. Zhang and K. Zhang. Improved stability for analytic quasi-convex nearly integrable systems and optimal speed of arnold diffusion. Nonlinearity, 30(7):2918, jun 2017

work page 2017

[36] [36]

K. Zhang. Speed of arnold diffusion for analytic hamiltonian systems. Inventiones Mathematicae, 186:255–290, 11 2011. EFFECTIVE STABILITY ESTIMATES CLOSE TO RESONANCES 39 Department of Mathematics, University of Roma Tor Vergata, Via della Ricerca Scientifica 1, 00133 Roma (Italy) Email address:celletti@mat.uniroma2.it Department of Mathematics, Universit...

work page 2011