# 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 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.
 [1] Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533 [2] Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643 [3] Xiaocai Wang, Junxiang Xu, Dongfeng Zhang. A KAM theorem for the elliptic lower dimensional tori with one normal frequency in reversible systems. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2141-2160. doi: 10.3934/dcds.2017092 [4] Luca Biasco, Luigi Chierchia. Exponential stability for the resonant D'Alembert model of celestial mechanics. Discrete and Continuous Dynamical Systems, 2005, 12 (4) : 569-594. doi: 10.3934/dcds.2005.12.569 [5] Luca Biasco, Luigi Chierchia. On the measure of KAM tori in two degrees of freedom. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6635-6648. doi: 10.3934/dcds.2020134 [6] Dongfeng Yan. KAM Tori for generalized Benjamin-Ono equation. Communications on Pure and Applied Analysis, 2015, 14 (3) : 941-957. doi: 10.3934/cpaa.2015.14.941 [7] M. I. Alomar, David Sánchez. Thermopower of a graphene monolayer with inhomogeneous spin-orbit interaction. Conference Publications, 2015, 2015 (special) : 1-9. doi: 10.3934/proc.2015.0001 [8] Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete and Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363 [9] Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete and Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 [10] Lorenzo Arona, Josep J. Masdemont. Computation of heteroclinic orbits between normally hyperbolic invariant 3-spheres foliated by 2-dimensional invariant Tori in Hill's problem. Conference Publications, 2007, 2007 (Special) : 64-74. doi: 10.3934/proc.2007.2007.64 [11] Guanghua Shi, Dongfeng Yan. KAM tori for quintic nonlinear schrödinger equations with given potential. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2421-2439. doi: 10.3934/dcds.2020120 [12] Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57 [13] Luigi Barletti, Philipp Holzinger, Ansgar Jüngel. Formal derivation of quantum drift-diffusion equations with spin-orbit interaction. Kinetic and Related Models, 2022, 15 (2) : 257-282. doi: 10.3934/krm.2022007 [14] Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362 [15] John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170 [16] Gabriela Jaramillo. Rotating spirals in oscillatory media with nonlocal interactions and their normal form. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022085 [17] Shengqing Hu, Bin Liu. Degenerate lower dimensional invariant tori in reversible system. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 3735-3763. doi: 10.3934/dcds.2018162 [18] Hsuan-Wen Su. Finding invariant tori with Poincare's map. Communications on Pure and Applied Analysis, 2008, 7 (2) : 433-443. doi: 10.3934/cpaa.2008.7.433 [19] Ugo Locatelli, Antonio Giorgilli. Invariant tori in the Sun--Jupiter--Saturn system. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 377-398. doi: 10.3934/dcdsb.2007.7.377 [20] Fuzhong Cong, Yong Li. Invariant hyperbolic tori for Hamiltonian systems with degeneracy. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 371-382. doi: 10.3934/dcds.1997.3.371

2021 Impact Factor: 1.497