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
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References:
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