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October  2020, 40(12): 6635-6648. doi: 10.3934/dcds.2020134

On the measure of KAM tori in two degrees of freedom

Luca Biasco and  Luigi Chierchia

Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Largo San L. Murialdo 1 - 00146 Roma, Italy

Received  August 2019 Revised  November 2019 Published  February 2020

A conjecture of Arnold, Kozlov and Neishtadt on the exponentially small measure of the "non-torus" set in analytic systems with two degrees of freedom is discussed.

Keywords: KAM theory, secondary tori, Hamiltonian systems, perturbation theory, mechanical systems.
Mathematics Subject Classification: 37J40, 70H08.
Citation: Luca Biasco, Luigi Chierchia. On the measure of KAM tori in two degrees of freedom. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6635-6648. doi: 10.3934/dcds.2020134
References:
[1]

V. I. Arnold, Conditions for the applicability, and estimate of the error, of an averaging method for systems which pass through states of resonance in the course of their evolution, Collected Works, 1 (1965), 477-480.  doi: 10.1007/978-3-642-01742-1_31.  Google Scholar

[2]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 2006.  Google Scholar

[3]

L. Biasco and L. Chierchia, On the measure of Lagrangian invariant tori in nearly–integrable mechanical systems, Rend. Lincei Mat. Appl., 26 (2015), 423-432.  doi: 10.4171/RLM/713.  Google Scholar

[4]

L. Biasco and L. Chierchia, KAM Theory for secondary tori, arXiv: 1702.06480v1 [math.DS]. Google Scholar

[5]

L. Biasco and L. Chierchia, Explicit estimates on the measure of primary KAM tori, Ann. Mat. Pura Appl. (4), 197 (2018), 261-281.  doi: 10.1007/s10231-017-0678-8.  Google Scholar

[6]

L. Biasco and L. Chierchia, On the topology of nearly–integrable Hamiltonians at simple resonances., To appear in Nonlinearity, 2020. arXiv: 1907.09434 [math.DS] Google Scholar

[7]

L. Biasco and L. Chierchia, Exponentially small measure of the non–torus set in 2 degrees of freedom, Work in progress. Google Scholar

[8]

P. A. M. Dirac, The adiabatic invariance of the quantum integrals, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 107 (1925), 725-734.  doi: 10.1098/rspa.1925.0052.  Google Scholar

[9]

M. Guzzo, L. Chierchia and G. Benettin, The steep Nekhoroshev Theorem, Commun. Math. Phys., 342 (2016), 569-601.  doi: 10.1007/s00220-015-2555-x.  Google Scholar

[10]

B.R. Hunt, T. Sauer and J.A. Yorke, Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 217-238.  doi: 10.1090/S0273-0979-1992-00328-2.  Google Scholar

[11]

B. R. Hunt and V. Y. Kaloshin, Prevalence, chapter 2, Handbook in Dynamical Systems, edited by H. Broer, F. Takens, B. Hasselblatt, 3 (2010), 43–87. Google Scholar

[12]

A. G. Medvedev, A. I. Neishtadt and D. V. Treschev, Lagrangian tori near resonances of near–integrable Hamiltonian systems, Nonlinearity, 28 (2015), 2105-2130.  doi: 10.1088/0951-7715/28/7/2105.  Google Scholar

[13]

A. I. Neishtadt, On passage through resonances in the two-frequency problem, Sov. Phys., Dokl., 20 (1975), 189-191.   Google Scholar

[14]

A. I. Neishtadt, Averaging, passage through resonances, and capture into resonance in two–frequency systems, Russian Math. Surveys, 69 (2014), 771-843.  doi: 10.4213/rm9603.  Google Scholar

[15]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly- integrable Hamiltonian systems I, Math. Surveys, 32 (1977), 1-65.   Google Scholar

[16]

J. Pöschel, Nekhoroshev estimates for quasi–convex Hamiltonian systems, Math. Z., 213 (1993), 187-216.  doi: 10.1007/BF03025718.  Google Scholar

show all references

References:
[1]

V. I. Arnold, Conditions for the applicability, and estimate of the error, of an averaging method for systems which pass through states of resonance in the course of their evolution, Collected Works, 1 (1965), 477-480.  doi: 10.1007/978-3-642-01742-1_31.  Google Scholar

[2]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 2006.  Google Scholar

[3]

L. Biasco and L. Chierchia, On the measure of Lagrangian invariant tori in nearly–integrable mechanical systems, Rend. Lincei Mat. Appl., 26 (2015), 423-432.  doi: 10.4171/RLM/713.  Google Scholar

[4]

L. Biasco and L. Chierchia, KAM Theory for secondary tori, arXiv: 1702.06480v1 [math.DS]. Google Scholar

[5]

L. Biasco and L. Chierchia, Explicit estimates on the measure of primary KAM tori, Ann. Mat. Pura Appl. (4), 197 (2018), 261-281.  doi: 10.1007/s10231-017-0678-8.  Google Scholar

[6]

L. Biasco and L. Chierchia, On the topology of nearly–integrable Hamiltonians at simple resonances., To appear in Nonlinearity, 2020. arXiv: 1907.09434 [math.DS] Google Scholar

[7]

L. Biasco and L. Chierchia, Exponentially small measure of the non–torus set in 2 degrees of freedom, Work in progress. Google Scholar

[8]

P. A. M. Dirac, The adiabatic invariance of the quantum integrals, Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci., 107 (1925), 725-734.  doi: 10.1098/rspa.1925.0052.  Google Scholar

[9]

M. Guzzo, L. Chierchia and G. Benettin, The steep Nekhoroshev Theorem, Commun. Math. Phys., 342 (2016), 569-601.  doi: 10.1007/s00220-015-2555-x.  Google Scholar

[10]

B.R. Hunt, T. Sauer and J.A. Yorke, Prevalence: a translation-invariant "almost every" on infinite-dimensional spaces, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 217-238.  doi: 10.1090/S0273-0979-1992-00328-2.  Google Scholar

[11]

B. R. Hunt and V. Y. Kaloshin, Prevalence, chapter 2, Handbook in Dynamical Systems, edited by H. Broer, F. Takens, B. Hasselblatt, 3 (2010), 43–87. Google Scholar

[12]

A. G. Medvedev, A. I. Neishtadt and D. V. Treschev, Lagrangian tori near resonances of near–integrable Hamiltonian systems, Nonlinearity, 28 (2015), 2105-2130.  doi: 10.1088/0951-7715/28/7/2105.  Google Scholar

[13]

A. I. Neishtadt, On passage through resonances in the two-frequency problem, Sov. Phys., Dokl., 20 (1975), 189-191.   Google Scholar

[14]

A. I. Neishtadt, Averaging, passage through resonances, and capture into resonance in two–frequency systems, Russian Math. Surveys, 69 (2014), 771-843.  doi: 10.4213/rm9603.  Google Scholar

[15]

N. N. Nekhoroshev, An exponential estimate of the time of stability of nearly- integrable Hamiltonian systems I, Math. Surveys, 32 (1977), 1-65.   Google Scholar

[16]

J. Pöschel, Nekhoroshev estimates for quasi–convex Hamiltonian systems, Math. Z., 213 (1993), 187-216.  doi: 10.1007/BF03025718.  Google Scholar

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