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November  2021, 41(11): 5165-5182. doi: 10.3934/dcds.2021072

Uniqueness properties of the KAM curve

IMJ-PRG, Université de Paris, Paris, France

Received  April 2020 Revised  March 2021 Published  November 2021 Early access  April 2021

Classical KAM theory guarantees the existence of a positive measure set of invariant tori for sufficiently smooth non-degenerate near-integrable systems. When seen as a function of the frequency this invariant collection of tori is called the KAM curve of the system. Restricted to analytic regularity, we obtain strong uniqueness properties for these objects. In particular, we prove that KAM curves completely characterize the underlying systems. We also show some of the dynamical implications on systems whose KAM curves share certain common features.

Citation: Frank Trujillo. Uniqueness properties of the KAM curve. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5165-5182. doi: 10.3934/dcds.2021072
References:
[1]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, 3, Springer Science & Business Media, 2007.

[2]

J.-B. T. Bost, Tores invariants des systèmes dynamiques hamiltoniens (d'après Kolmogorov, Arnol'd, Moser, Rüssmann, Zehnder, Herman, Pöschel), Astérisque, 113–157.

[3]

A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-540-45330-7.

[4]

C. Carminati, S. Marmi and D. Sauzin, There is only one KAM curve, Nonlinearity, 27 (2014), 2035. doi: 10.1088/0951-7715/27/9/2035.

[5]

L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals——elliptic case, Commentarii Mathematici Helvetici, 65 (1990), 4-35.  doi: 10.1007/BF02566590.

[6]

L. H. EliassonB. Fayad and R. Krikorian, Around the stability of KAM tori, Duke Mathematical Journal, 164 (2015), 1733-1775.  doi: 10.1215/00127094-3120060.

[7]

V. F. Lazutkin, Existence of caustics for the billiard problem in a convex domain, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 37 (1973), 186-216. 

[8]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Communications on Pure and Applied Mathematics, 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.

[9]

Z.-J. Shang, A Note on the KAM Theorem for Symplectic Mappings, Journal of Dynamics and Differential Equations, 12 (2000), 357-383.  doi: 10.1023/A:1009068425415.

show all references

References:
[1]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, 3, Springer Science & Business Media, 2007.

[2]

J.-B. T. Bost, Tores invariants des systèmes dynamiques hamiltoniens (d'après Kolmogorov, Arnol'd, Moser, Rüssmann, Zehnder, Herman, Pöschel), Astérisque, 113–157.

[3]

A. Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-540-45330-7.

[4]

C. Carminati, S. Marmi and D. Sauzin, There is only one KAM curve, Nonlinearity, 27 (2014), 2035. doi: 10.1088/0951-7715/27/9/2035.

[5]

L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals——elliptic case, Commentarii Mathematici Helvetici, 65 (1990), 4-35.  doi: 10.1007/BF02566590.

[6]

L. H. EliassonB. Fayad and R. Krikorian, Around the stability of KAM tori, Duke Mathematical Journal, 164 (2015), 1733-1775.  doi: 10.1215/00127094-3120060.

[7]

V. F. Lazutkin, Existence of caustics for the billiard problem in a convex domain, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 37 (1973), 186-216. 

[8]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Communications on Pure and Applied Mathematics, 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.

[9]

Z.-J. Shang, A Note on the KAM Theorem for Symplectic Mappings, Journal of Dynamics and Differential Equations, 12 (2000), 357-383.  doi: 10.1023/A:1009068425415.

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