# American Institute of Mathematical Sciences

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. [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. [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. [4] A. Celletti, Periodic and quasi-periodic attractors of weakly-dissipative nearly-integrable systems,, Reg. Ch. Dyn., 14 (2009), 49. doi: 10.1134/S1560354709010067. [5] A. Celletti and L. Chierchia, A constructive theory of Lagrangian tori and computer-assisted applications,, in, 4 (1995), 60. [6] A. Celletti and L. Chierchia, KAM stability and celestial mechanics,, Memoirs American Mathematical Society, 187 (2007). [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. [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. [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. [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. [11] R. de la Llave, A. González, À. Jorba and J. Villanueva, KAM theory without action-angle variables,, Nonlinearity, 18 (2005), 855. [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). [13] J.-P. Eckmann and P. Wittwer, "Computer methods and Borel summability Applied to Feigenbaum's Equation,", Lecture Notes in Physics, 227 (1985). [14] A. Giorgilli, Quantitative methods in classical perturbation theory,, in, 336 (1993). [15] A. Giorgilli, Notes on exponential stability of Hamiltonian systems,, in, (2003), 87. [16] A. Giorgilli, Sistemi Dinamici II,, Lecture Notes for Students, (2010). [17] A. Giorgilli and U. Locatelli, On classical series expansion for quasi-periodic motions,, MPEJ, 3 (1997), 1. [18] A. Giorgilli and S. Marmi, Convergence radius in the Poincaré-Siegel problem,, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 601. [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. [20] O. E. Lanford, III, A computer-assisted proof of the Feigenbaum conjectures,, Bull. of the Amer. Math. Soc. (N.S.), 6 (1982), 427. [21] J. Laskar, Introduction to frequency map analysis,, in, 533 (1999), 134. [22] J. Laskar, Frequency Map analysis and quasi periodic decompositions,, in, (2005), 99. [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. [24] J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. G\, 1962 (1962), 1. [25] M. B. Sevryuk, "Reversible Systems,", Lect. Notes Math., 1211 (1211). [26] L. Stefanelli, "Periodic and Quasi-Periodic Motions in Nearly-Integrable Dissipative Systems with Application to Celestial Mechanics,", Ph.D. Thesis, (2011).

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##### 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. [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. [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. [4] A. Celletti, Periodic and quasi-periodic attractors of weakly-dissipative nearly-integrable systems,, Reg. Ch. Dyn., 14 (2009), 49. doi: 10.1134/S1560354709010067. [5] A. Celletti and L. Chierchia, A constructive theory of Lagrangian tori and computer-assisted applications,, in, 4 (1995), 60. [6] A. Celletti and L. Chierchia, KAM stability and celestial mechanics,, Memoirs American Mathematical Society, 187 (2007). [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. [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. [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. [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. [11] R. de la Llave, A. González, À. Jorba and J. Villanueva, KAM theory without action-angle variables,, Nonlinearity, 18 (2005), 855. [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). [13] J.-P. Eckmann and P. Wittwer, "Computer methods and Borel summability Applied to Feigenbaum's Equation,", Lecture Notes in Physics, 227 (1985). [14] A. Giorgilli, Quantitative methods in classical perturbation theory,, in, 336 (1993). [15] A. Giorgilli, Notes on exponential stability of Hamiltonian systems,, in, (2003), 87. [16] A. Giorgilli, Sistemi Dinamici II,, Lecture Notes for Students, (2010). [17] A. Giorgilli and U. Locatelli, On classical series expansion for quasi-periodic motions,, MPEJ, 3 (1997), 1. [18] A. Giorgilli and S. Marmi, Convergence radius in the Poincaré-Siegel problem,, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 601. [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. [20] O. E. Lanford, III, A computer-assisted proof of the Feigenbaum conjectures,, Bull. of the Amer. Math. Soc. (N.S.), 6 (1982), 427. [21] J. Laskar, Introduction to frequency map analysis,, in, 533 (1999), 134. [22] J. Laskar, Frequency Map analysis and quasi periodic decompositions,, in, (2005), 99. [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. [24] J. Moser, On invariant curves of area-preserving mappings of an annulus,, Nachr. Akad. Wiss. G\, 1962 (1962), 1. [25] M. B. Sevryuk, "Reversible Systems,", Lect. Notes Math., 1211 (1211). [26] L. Stefanelli, "Periodic and Quasi-Periodic Motions in Nearly-Integrable Dissipative Systems with Application to Celestial Mechanics,", Ph.D. Thesis, (2011).
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