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 and 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), 13-40; Russ. Math. Surv., 18 (1963).

[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), 201-223. 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), 667-770. 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), 49-63. doi: 10.1134/S1560354709010067.

[5]

A. Celletti and L. Chierchia, A constructive theory of Lagrangian tori and computer-assisted applications, in "Dynamics Reported," (eds. C. K. R. T. Jones, U. Kirchgraber and H. O. Walther), Dynam. Report. Expositions Dynam. Systems (N.S.), 4, Springer, Berlin, (1995), 60-129.

[6]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Memoirs American Mathematical Society, 187 (2007), viii+134 pp.

[7]

A. Celletti and L. Chierchia, Measures of basins of attraction in spin-orbit dynamics, Cel. Mech. Dyn. Astr., 101 (2008), 159-170. 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), 311-345. 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), 151-171. 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), 397-412. 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), 855-895.

[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, Graduate Texts in Mathematics, 93, Springer-Verlag, New York, 1992.

[13]

J.-P. Eckmann and P. Wittwer, "Computer methods and Borel summability Applied to Feigenbaum's Equation," Lecture Notes in Physics, 227, Springer-Verlag, Berlin, 1985.

[14]

A. Giorgilli, Quantitative methods in classical perturbation theory, in "From Newton to Chaos'' (eds. A. E. Roy and B. D. Steves) (Cortina d'Ampezzo, 1993), NATO Adv. Sci. Inst. Ser. B Phys., 336, Plenum, New York, 1995.

[15]

A. Giorgilli, Notes on exponential stability of Hamiltonian systems, in "Dynamical Systems. Part I,'' Pubblicazioni del Centro di Ricerca Matematica Ennio De Giorgi, Scuola Norm. Sup., Pisa, (2003), 87-198.

[16]

A. Giorgilli, Sistemi Dinamici II, Lecture Notes for Students, 2010. Available from: http://newrobin.mat.unimi.it/users/antonio/sisdin/sisdin.html.

[17]

A. Giorgilli and U. Locatelli, On classical series expansion for quasi-periodic motions, MPEJ, 3 (1997), 1-25.

[18]

A. Giorgilli and S. Marmi, Convergence radius in the Poincaré-Siegel problem, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 601-621.

[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), 527-530; Engl. transl. in Los Alamos Scientific Laboratory translation LA-TR-71-67; reprinted in Lecture Notes in Physics, 93.

[20]

O. E. Lanford, III, A computer-assisted proof of the Feigenbaum conjectures, Bull. of the Amer. Math. Soc. (N.S.), 6 (1982), 427-434.

[21]

J. Laskar, Introduction to frequency map analysis, in "Hamiltonian Systems with Three or More Degrees of Freedom'' (ed. C. Simò) (S'Agaró, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, (1999), 134-150.

[22]

J. Laskar, Frequency Map analysis and quasi periodic decompositions, in "Hamiltonian Systems and Fourier Analysis,'' Adv. Astron. Astrophys., Camb. Sci. Publ., Cambridge, (2005), 99-133.

[23]

U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems, Cel. Mech. & Dyn. Astr., 78 (2000), 47-74. doi: 10.1023/A:1011139523256.

[24]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött. Math.-Phys. Kl. II, 1962 (1962), 1-20.

[25]

M. B. Sevryuk, "Reversible Systems," Lect. Notes Math., 1211, Springer-Verlag, Berlin, 1986.

[26]

L. Stefanelli, "Periodic and Quasi-Periodic Motions in Nearly-Integrable Dissipative Systems with Application to Celestial Mechanics," Ph.D. Thesis, Univ. Roma "Tor Vergata,'' 2011.

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), 13-40; Russ. Math. Surv., 18 (1963).

[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), 201-223. 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), 667-770. 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), 49-63. doi: 10.1134/S1560354709010067.

[5]

A. Celletti and L. Chierchia, A constructive theory of Lagrangian tori and computer-assisted applications, in "Dynamics Reported," (eds. C. K. R. T. Jones, U. Kirchgraber and H. O. Walther), Dynam. Report. Expositions Dynam. Systems (N.S.), 4, Springer, Berlin, (1995), 60-129.

[6]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Memoirs American Mathematical Society, 187 (2007), viii+134 pp.

[7]

A. Celletti and L. Chierchia, Measures of basins of attraction in spin-orbit dynamics, Cel. Mech. Dyn. Astr., 101 (2008), 159-170. 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), 311-345. 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), 151-171. 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), 397-412. 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), 855-895.

[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, Graduate Texts in Mathematics, 93, Springer-Verlag, New York, 1992.

[13]

J.-P. Eckmann and P. Wittwer, "Computer methods and Borel summability Applied to Feigenbaum's Equation," Lecture Notes in Physics, 227, Springer-Verlag, Berlin, 1985.

[14]

A. Giorgilli, Quantitative methods in classical perturbation theory, in "From Newton to Chaos'' (eds. A. E. Roy and B. D. Steves) (Cortina d'Ampezzo, 1993), NATO Adv. Sci. Inst. Ser. B Phys., 336, Plenum, New York, 1995.

[15]

A. Giorgilli, Notes on exponential stability of Hamiltonian systems, in "Dynamical Systems. Part I,'' Pubblicazioni del Centro di Ricerca Matematica Ennio De Giorgi, Scuola Norm. Sup., Pisa, (2003), 87-198.

[16]

A. Giorgilli, Sistemi Dinamici II, Lecture Notes for Students, 2010. Available from: http://newrobin.mat.unimi.it/users/antonio/sisdin/sisdin.html.

[17]

A. Giorgilli and U. Locatelli, On classical series expansion for quasi-periodic motions, MPEJ, 3 (1997), 1-25.

[18]

A. Giorgilli and S. Marmi, Convergence radius in the Poincaré-Siegel problem, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 601-621.

[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), 527-530; Engl. transl. in Los Alamos Scientific Laboratory translation LA-TR-71-67; reprinted in Lecture Notes in Physics, 93.

[20]

O. E. Lanford, III, A computer-assisted proof of the Feigenbaum conjectures, Bull. of the Amer. Math. Soc. (N.S.), 6 (1982), 427-434.

[21]

J. Laskar, Introduction to frequency map analysis, in "Hamiltonian Systems with Three or More Degrees of Freedom'' (ed. C. Simò) (S'Agaró, 1995), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 533, Kluwer Acad. Publ., Dordrecht, (1999), 134-150.

[22]

J. Laskar, Frequency Map analysis and quasi periodic decompositions, in "Hamiltonian Systems and Fourier Analysis,'' Adv. Astron. Astrophys., Camb. Sci. Publ., Cambridge, (2005), 99-133.

[23]

U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems, Cel. Mech. & Dyn. Astr., 78 (2000), 47-74. doi: 10.1023/A:1011139523256.

[24]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött. Math.-Phys. Kl. II, 1962 (1962), 1-20.

[25]

M. B. Sevryuk, "Reversible Systems," Lect. Notes Math., 1211, Springer-Verlag, Berlin, 1986.

[26]

L. Stefanelli, "Periodic and Quasi-Periodic Motions in Nearly-Integrable Dissipative Systems with Application to Celestial Mechanics," Ph.D. Thesis, Univ. Roma "Tor Vergata,'' 2011.

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