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June  2015, 20(4): 1155-1187. doi: 10.3934/dcdsb.2015.20.1155

Quasi-periodic motions in a special class of dynamical equations with dissipative effects: A pair of detection methods

 1 Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133 Roma 2 Geoazur, Université de Nice Sophia-Antipolis, Observatoire de la Côte d’Azur, 250, rue Albert Einstein, 06560 Valbonne, France

Received  July 2014 Revised  October 2014 Published  February 2015

We consider a particular class of equations of motion, generalizing to $n$ degrees of freedom the dissipative spin--orbit problem'',   commonly studied in Celestial Mechanics. Those equations are formulated in a pseudo-Hamiltonian framework with action-angle coordinates; they contain a quasi-integrable conservative part and friction terms, assumed to be linear and isotropic with respect to the action variables. In such a context, we transfer two methods determining quasi-periodic solutions, which were originally designed to analyze purely Hamiltonian quasi-integrable problems.
First, we show how the frequency map analysis can be adapted to this kind of dissipative models. Our approach is based on a key remark: the method can work as usual, by studying the behavior of the angular velocities of the motions as a function of the so called external frequencies'', instead of the actions.
Moreover, we explicitly implement the Kolmogorov's normalization algorithm for the dissipative systems considered here. In a previous article, we proved a theoretical result: such a constructing procedure is convergent under the hypotheses usually assumed in KAM theory. In the present work, we show that it can be translated to a code making algebraic manipulations on a computer, so to calculate effectively quasi-periodic solutions on invariant tori (and the attracting dynamics in their neighborhoods).
Both the methods are carefully tested, by checking that their predictions are in agreement, in the case of the so called dissipative forced pendulum''. Furthermore, the results obtained by applying our adaptation of the frequency analysis method to the dissipative standard map are compared with some existing ones in the literature.
Citation: Ugo Locatelli, Letizia Stefanelli. Quasi-periodic motions in a special class of dynamical equations with dissipative effects: A pair of detection methods. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1155-1187. doi: 10.3934/dcdsb.2015.20.1155
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