October  2012, 17(7): 2561-2593. doi: 10.3934/dcdsb.2012.17.2561

Kolmogorov's normal form for equations of motion with dissipative effects

1. 

Geoazur, Université de Nice Sophia-Antipolis, Centre National de la Recherche Scientifique (UMR7329), Observatoire de la Côte d’Azur, Avenue Nicolas Copernic, 06130 Grasse, France

2. 

Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133 Roma

Received  July 2011 Revised  April 2012 Published  July 2012

We focus on the equations of motion related to the “dissipative spin–orbit model”, which is commonly studied in Celestial Mechanics. We consider them in the more general framework of a 2$n$–dimensional action–angle phase space. Since the friction terms are assumed to be linear and isotropic with respect to the action variables, the Kolmogorov’s normalization algorithm for quasi-integrable Hamiltonians can be easily adapted to the dissipative system considered here. This allows us to prove the existence of quasi-periodic invariant tori that are local attractors.
Citation: Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561
References:
[1]

V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under small perturbations of the Hamiltonian,, Usp. Mat. Nauk, 18 (1963), 13. Google Scholar

[2]

G. Benettin, L. Galgani, A. Giorgilli and J.-M. Strelcyn, A proof of Kolmogorov's theorem on invariant tori using canonical transformations defined by the Lie method,, Nuovo Cimento B (11), 79 (1984), 201. doi: 10.1007/BF02748972. Google Scholar

[3]

H. Broer, C. Simò and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms,, Nonlinearity, 11 (1998), 667. doi: 10.1088/0951-7715/11/3/015. Google Scholar

[4]

A. Celletti, Periodic and quasi-periodic attractors of weakly-dissipative nearly-integrable systems,, Reg. Ch. Dyn., 14 (2009), 49. doi: 10.1134/S1560354709010067. Google Scholar

[5]

A. Celletti and L. Chierchia, A constructive theory of Lagrangian tori and computer-assisted applications,, in, 4 (1995), 60. Google Scholar

[6]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics,, Memoirs American Mathematical Society, 187 (2007). Google Scholar

[7]

A. Celletti and L. Chierchia, Measures of basins of attraction in spin-orbit dynamics,, Cel. Mech. Dyn. Astr., 101 (2008), 159. doi: 10.1007/s10569-008-9142-9. Google Scholar

[8]

A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics, Arch. Rat. Mech. Anal., 191 (2009), 311. doi: 10.1007/s00205-008-0141-5. Google Scholar

[9]

A. Celletti and S. Di Ruzza, Periodic and quasi-periodic orbits of the dissipative standard map,, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 151. doi: 10.3934/dcdsb.2011.16.151. Google Scholar

[10]

A. Celletti, A. Giorgilli and U. Locatelli, Improved estimates on the existence of invariant tori for Hamiltonian systems,, Nonlinearity, 13 (2000), 397. doi: 10.1088/0951-7715/13/2/304. Google Scholar

[11]

R. de la Llave, A. González, À. Jorba and J. Villanueva, KAM theory without action-angle variables,, Nonlinearity, 18 (2005), 855. Google Scholar

[12]

B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, "Modern Geometry-Methods and Applications. Part I: The Geometry of Surfaces, Transformation Groups, and Fields,'', Second edition, 93 (1992). Google Scholar

[13]

J.-P. Eckmann and P. Wittwer, "Computer methods and Borel summability Applied to Feigenbaum's Equation,", Lecture Notes in Physics, 227 (1985). Google Scholar

[14]

A. Giorgilli, Quantitative methods in classical perturbation theory,, in, 336 (1993). Google Scholar

[15]

A. Giorgilli, Notes on exponential stability of Hamiltonian systems,, in, (2003), 87. Google Scholar

[16]

A. Giorgilli, Sistemi Dinamici II,, Lecture Notes for Students, (2010). Google Scholar

[17]

A. Giorgilli and U. Locatelli, On classical series expansion for quasi-periodic motions,, MPEJ, 3 (1997), 1. Google Scholar

[18]

A. Giorgilli and S. Marmi, Convergence radius in the Poincaré-Siegel problem,, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 601. Google Scholar

[19]

A. N. Kolmogorov, On Conservation of conditionally periodic movements with small change in the Hamilton function,, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527. Google Scholar

[20]

O. E. Lanford, III, A computer-assisted proof of the Feigenbaum conjectures,, Bull. of the Amer. Math. Soc. (N.S.), 6 (1982), 427. Google Scholar

[21]

J. Laskar, Introduction to frequency map analysis,, in, 533 (1999), 134. Google Scholar

[22]

J. Laskar, Frequency Map analysis and quasi periodic decompositions,, in, (2005), 99. Google Scholar

[23]

U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems,, Cel. Mech. & Dyn. Astr., 78 (2000), 47. doi: 10.1023/A:1011139523256. Google Scholar

[24]

J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. G\, 1962 (1962), 1. Google Scholar

[25]

M. B. Sevryuk, "Reversible Systems,", Lect. Notes Math., 1211 (1211). Google Scholar

[26]

L. Stefanelli, "Periodic and Quasi-Periodic Motions in Nearly-Integrable Dissipative Systems with Application to Celestial Mechanics,", Ph.D. Thesis, (2011). Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under small perturbations of the Hamiltonian,, Usp. Mat. Nauk, 18 (1963), 13. Google Scholar

[2]

G. Benettin, L. Galgani, A. Giorgilli and J.-M. Strelcyn, A proof of Kolmogorov's theorem on invariant tori using canonical transformations defined by the Lie method,, Nuovo Cimento B (11), 79 (1984), 201. doi: 10.1007/BF02748972. Google Scholar

[3]

H. Broer, C. Simò and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms,, Nonlinearity, 11 (1998), 667. doi: 10.1088/0951-7715/11/3/015. Google Scholar

[4]

A. Celletti, Periodic and quasi-periodic attractors of weakly-dissipative nearly-integrable systems,, Reg. Ch. Dyn., 14 (2009), 49. doi: 10.1134/S1560354709010067. Google Scholar

[5]

A. Celletti and L. Chierchia, A constructive theory of Lagrangian tori and computer-assisted applications,, in, 4 (1995), 60. Google Scholar

[6]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics,, Memoirs American Mathematical Society, 187 (2007). Google Scholar

[7]

A. Celletti and L. Chierchia, Measures of basins of attraction in spin-orbit dynamics,, Cel. Mech. Dyn. Astr., 101 (2008), 159. doi: 10.1007/s10569-008-9142-9. Google Scholar

[8]

A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics, Arch. Rat. Mech. Anal., 191 (2009), 311. doi: 10.1007/s00205-008-0141-5. Google Scholar

[9]

A. Celletti and S. Di Ruzza, Periodic and quasi-periodic orbits of the dissipative standard map,, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 151. doi: 10.3934/dcdsb.2011.16.151. Google Scholar

[10]

A. Celletti, A. Giorgilli and U. Locatelli, Improved estimates on the existence of invariant tori for Hamiltonian systems,, Nonlinearity, 13 (2000), 397. doi: 10.1088/0951-7715/13/2/304. Google Scholar

[11]

R. de la Llave, A. González, À. Jorba and J. Villanueva, KAM theory without action-angle variables,, Nonlinearity, 18 (2005), 855. Google Scholar

[12]

B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, "Modern Geometry-Methods and Applications. Part I: The Geometry of Surfaces, Transformation Groups, and Fields,'', Second edition, 93 (1992). Google Scholar

[13]

J.-P. Eckmann and P. Wittwer, "Computer methods and Borel summability Applied to Feigenbaum's Equation,", Lecture Notes in Physics, 227 (1985). Google Scholar

[14]

A. Giorgilli, Quantitative methods in classical perturbation theory,, in, 336 (1993). Google Scholar

[15]

A. Giorgilli, Notes on exponential stability of Hamiltonian systems,, in, (2003), 87. Google Scholar

[16]

A. Giorgilli, Sistemi Dinamici II,, Lecture Notes for Students, (2010). Google Scholar

[17]

A. Giorgilli and U. Locatelli, On classical series expansion for quasi-periodic motions,, MPEJ, 3 (1997), 1. Google Scholar

[18]

A. Giorgilli and S. Marmi, Convergence radius in the Poincaré-Siegel problem,, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 601. Google Scholar

[19]

A. N. Kolmogorov, On Conservation of conditionally periodic movements with small change in the Hamilton function,, Dokl. Akad. Nauk SSSR (N.S.), 98 (1954), 527. Google Scholar

[20]

O. E. Lanford, III, A computer-assisted proof of the Feigenbaum conjectures,, Bull. of the Amer. Math. Soc. (N.S.), 6 (1982), 427. Google Scholar

[21]

J. Laskar, Introduction to frequency map analysis,, in, 533 (1999), 134. Google Scholar

[22]

J. Laskar, Frequency Map analysis and quasi periodic decompositions,, in, (2005), 99. Google Scholar

[23]

U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems,, Cel. Mech. & Dyn. Astr., 78 (2000), 47. doi: 10.1023/A:1011139523256. Google Scholar

[24]

J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. G\, 1962 (1962), 1. Google Scholar

[25]

M. B. Sevryuk, "Reversible Systems,", Lect. Notes Math., 1211 (1211). Google Scholar

[26]

L. Stefanelli, "Periodic and Quasi-Periodic Motions in Nearly-Integrable Dissipative Systems with Application to Celestial Mechanics,", Ph.D. Thesis, (2011). Google Scholar

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