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

Ugo Locatelli; Letizia Stefanelli

doi:10.3934/dcdsb.2015.20.1155

Article Contents

2015, Volume 20, Issue 4: 1155-1187. Doi: 10.3934/dcdsb.2015.20.1155

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

Ugo Locatelli^1, and
Letizia Stefanelli^2,

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

Received: June 30, 2014

Revised: September 30, 2014

Published: May 2015

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Abstract

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.

Keywords:

Frequency analysis,
normal form methods,
KAM theory,
attractors,
dissipative spin--orbit problem in Celestial Mechanics,
numerical and semi-analytic methods in dynamical systems.

Mathematics Subject Classification: Primary: 34C20; Secondary: 34D10, 37J40, 70F15, 70F40.

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References

[1]	A. Abad, R. Barrio, F. Blesa and M. Rodriguez, Algorithm 924: TIDES, a Taylor Series Integrator for Differential EquationS, ACM Transactions on Math. Software, 39 (2012), Art. 5, 28 pp.doi: 10.1145/2382585.2382590.
[2]	V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, Usp. Mat. Nauk, 18 (1963), 13-40.
[3]	D. Bambusi and E. Haus, Asymptotic stability of synchronous orbits for a gravitating viscoelastic sphere, Cel. Mech. & Dyn. Astr., 114 (2012), 255-277.doi: 10.1007/s10569-012-9438-7.
[4]	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, 79 (1984), 201-223.doi: 10.1007/BF02748972.
[5]	L. Biasco and L. Chierchia, Low-order resonances in weakly dissipative spin-orbit models, J. Diff. Equations, 246 (2009), 4345-4370.doi: 10.1016/j.jde.2008.11.008.
[6]	H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-periodic Motions in Families of Dynamical Systems. Order Amidst Chaos, Lecture Notes in Mathematics, 1645, Springer-Verlag, Berlin, 1996.
[7]	H. W. 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.
[8]	R. Calleja and A. Celletti, Breakdown of invariant attractors for the dissipative standard map, CHAOS, 20 (2010), 013121, 9pp.doi: 10.1063/1.3335408.
[9]	R. Calleja, A. Celletti and R. de la Llave, A KAM theory for conformally symplectic systems: Efficient algorithms and their validation, J. Diff. Equations, 255 (2013), 978-1049.doi: 10.1016/j.jde.2013.05.001.
[10]	R. Calleja, A. Celletti and R. de la Llave, Local behavior near quasi-periodic solutions of conformally symplectic systems, J. Dyn. & Diff. Equations, 25 (2013), 821-841.doi: 10.1007/s10884-013-9319-0.
[11]	R. Calleja and R. de la Llave, A numerically accessible criterion for the breakdown of quasi-periodic solutions and its rigorous justification, Nonlinearity, 23 (2010), 2029-2058.doi: 10.1088/0951-7715/23/9/001.
[12]	A. Celletti, Analysis of resonances in the spin-orbit problem in Celestial Mechanics: The synchronous resonance (Part I), J. of App. Math. and Phys. (ZAMP), 41 (1990), 174-204.doi: 10.1007/BF00945107.
[13]	A. Celletti, Analysis of resonances in the spin-orbit problem in Celestial Mechanics: higher order resonances and some numerical experiments (Part II), J. of App. Math. and Phys. (ZAMP), 41 (1990), 453-479.doi: 10.1007/BF00945951.
[14]	A. Celletti, Periodic and quasi-periodic attractors of weakly-dissipative nearly-integrable systems, Reg. & Ch. Dyn., 14 (2009), 49-63.doi: 10.1134/S1560354709010067.
[15]	A. Celletti, Stability and Chaos in Celestial Mechanics, Springer-Praxis, 2010.doi: 10.1007/978-3-540-85146-2.
[16]	A. Celletti and L. Chierchia, KAM stability and celestial mechanics, Memoirs American Mathematical Society, 187 (2007), viii+134 pp.doi: 10.1090/memo/0878.
[17]	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.
[18]	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.
[19]	A. Celletti and S. Di Ruzza, Periodic and quasi-periodic orbits of the dissipative standard map, DCDS-B, 16 (2011), 151-171.doi: 10.3934/dcdsb.2011.16.151.
[20]	A. Celletti, S. Di Ruzza, C. Lhotka and L. Stefanelli, Nearly-integrable dissipative systems and celestial mechanics, The European Phys. Jour. - Special Topics, 186 (2010), 33-66.doi: 10.1140/epjst/e2010-01259-2.
[21]	A. Celletti, C. Froeschlé and E. Lega, Dissipative and weakly-dissipative regimes in nearly-integrable mappings, DCDS-A, 16 (2006), 757-781.doi: 10.3934/dcds.2006.16.757.
[22]	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.
[23]	C. Chandre, J. Laskar, G. Benfatto and H. R. Jauslin, Determination of the breakup of invariant tori in three frequency Hamiltonian systems, Physica D, 154 (2001), 159-170.doi: 10.1016/S0167-2789(01)00268-8.
[24]	L. Chierchia, A. N. Kolmogorov's 1954 paper on nearly-integrable Hamiltonian systems, Reg. & Ch. Dyn., 13 (2008), 130-139.doi: 10.1134/S1560354708020056.
[25]	A. C. M. Correia and J. Laskar, Mercury's capture into the 3/2 spin-orbit resonance as a result of its chaotic dynamics, Nature, 429 (2004), 848-850.doi: 10.1038/nature02609.
[26]	A. Deprit and A. Deprit-Bartholomé, Stability of the triangular Lagrangian points, Astron. J., 72 (1967), p173.doi: 10.1086/110213.
[27]	S. D'Hoedt and A. Lemaître, Planetary long periodic terms in Mercury's rotation: A two dimensional adiabatic approach, Cel. Mech. & Dyn. Astr., 101 (2008), 127-139.
[28]	S. Dumas and J. Laskar, Global Dynamics and Long-Time Stability in Hamiltonian Systems via Numerical Frequency Analysis, Phys. Rev. Lett., 70 (1993), 2975-2979.doi: 10.1103/PhysRevLett.70.2975.
[29]	F. Gabern, A. Jorba and U. Locatelli, On the construction of the Kolmogorov normal form for the Trojan asteroids, Nonlinearity, 18 (2005), 1705-1734.doi: 10.1088/0951-7715/18/4/017.
[30]	C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.
[31]	A. Giorgilli and U. Locatelli, Kolmogorov theorem and classical perturbation theory, J. of App. Math. and Phys. (ZAMP), 48 (1997), 220-261.doi: 10.1007/PL00001475.
[32]	A. Giorgilli, U. Locatelli and M. Sansottera, Kolmogorov and Nekhoroshev theory for the problem of three bodies, Cel. Mech. & Dyn. Astr., 104 (2009), 159-173.doi: 10.1007/s10569-009-9192-7.
[33]	A. Giorgilli and M. Sansottera, Methods of algebraic manipulation in perturbation theory, in Chaos, Diffusion and Non-integrability in Hamiltonian Systems - Applications to Astronomy, Proceedings of the 3rd La Plata International School on Astronomy and Geophysics, (eds. P.M. Cincotta, C.M. Giordano and C. Efthymiopoulos), Universidad Nacional de La Plata and Asociación Argentina de Astronomía Publishers, La Plata, 2012.
[34]	P. Goldreich and S. J. Peale, Spin-orbit coupling in the Solar System, Astron. J., 71 (1966), 425-438.doi: 10.1086/109947.
[35]	P. Goldreich and S. J. Peale, The dynamics of planetary rotations, Ann. Rev. Astron. Astrophys., 6 (1968), 287-320.doi: 10.1146/annurev.aa.06.090168.001443.
[36]	G. Gomez, J. M. Mondelo and C. Simò, A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples, DCDS-B, 14 (2010), 41-74.doi: 10.3934/dcdsb.2010.14.41.
[37]	G. Gomez, J. M. Mondelo and C. Simò, A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates, DCDS-B, 14 (2010), 75-109.doi: 10.3934/dcdsb.2010.14.75.
[38]	M. Govin, C. Chandre and H. R. Jauslin, KAM-Renormalization-Group analysis of stability in Hamiltonian flows, Phys. Rev. Lett., 79 (1997), 3881-3884.
[39]	J. M. Greene, A method for determining a stochastic transition, J. of Math. Phys, 20 (1979), 1183-1201.doi: 10.1063/1.524170.
[40]	E. Haus and D. Bambusi, Asymptotic behavior of an elastic satellite with internal friction, Celestial Mechanics and Dynamical Astronomy, 114 (2012), 255-277, arXiv:1212.0816.doi: 10.1007/s10569-012-9438-7.
[41]	M. Hénon, Exploration numérique du problème restreint IV: Masses égales, orbites non périodiques, Bulletin Astronomique, 3 (1966), 49-66.
[42]	A. Jorba and M. Zou, A Software Package for the Numerical Integration of ODEs by Means of High-Order Taylor Methods, Experiment. Math., 14 (2005), 99-117.doi: 10.1080/10586458.2005.10128904.
[43]	A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function, Dokl. Akad. Nauk SSSR, 98 (1954), 527-530; Engl. transl. in: Los Alamos Scientific Laboratory translation LA-TR-71-67; reprinted in: Lecture Notes in Physics 93.
[44]	J. Laskar, Introduction to frequency map analysis, in Hamiltonian Systems with Three or More Degrees of Freedom (ed. C. Simò), Proceedings of the NATO ASI school held in S'Agaro (Spain, June 19-30, 1995), Kluwer, 533 (1999), 134-150.
[45]	J. Laskar, Frequency Map analysis and quasi periodic decompositions, in Hamiltonian systems and Fourier analysis (eds. Benest et al.), Taylor and Francis, 2005.
[46]	J. Laskar, C. Froeschlé and A. Celletti, The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping, Physica D, 56 (1992), 253-269.doi: 10.1016/0167-2789(92)90028-L.
[47]	J. Laskar and P. Robutel, The chaotic obliquity of the planets, Nature, 361 (1993), 608-612.doi: 10.1038/361608a0.
[48]	E. Lega and C. Froeschlé, Numerical investigations of the structure around an invariant KAM torus using the frequency map analysis, Physica D, 95 (1996), 97-106.doi: 10.1016/0167-2789(96)00046-2.
[49]	A. M. Leontovich, On the stability of the Lagrange periodic solutions for the reduced problem of three bodies, Soviet Math. Dokl., 3 (1962), p425.
[50]	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.
[51]	U. Locatelli and A. Giorgilli, Construction of the Kolmogorov's normal form for a planetary system, Reg. & Ch. Dyn., 10 (2005), 153-171.doi: 10.1070/RD2005v010n02ABEH000309.
[52]	U. Locatelli and A. Giorgilli, Invariant tori in the Sun-Jupiter-Saturn system, DCDS-B, 7 (2007), 377-398.doi: 10.3934/dcdsb.2007.7.377.
[53]	G. J. F. MacDonald, Tidal friction, Rev. Geophys., 2 (1964), 467-541.doi: 10.1029/RG002i003p00467.
[54]	R. S. MacKay, Greene's residue criterion, Nonlinearity, 5 (1992), 161-187.doi: 10.1088/0951-7715/5/1/007.
[55]	A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori, J. Stat. Phys., 78 (1995), 1607-1617.doi: 10.1007/BF02180145.
[56]	J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött., II Math. Phys. Kl, 1962 (1962), 1-20.
[57]	A. Noullez, Chaos characterization in Hamiltonian systems using resonance analysis, in Dynamics of Celestial Bodies - DCB-08 International Conference Proceedings, (2009), 147-150.
[58]	Y. Papaphilippou and J. Laskar, Global dynamics of triaxial galactic models through frequency map analysis, Astron. & Astrophys., 329 (1998), 451-481.
[59]	S. J. Peale, The free precession and libration of Mercury, Icarus, 178 (2005), 4-18.doi: 10.1016/j.icarus.2005.03.017.
[60]	J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.doi: 10.1002/cpa.3160350504.
[61]	P. Robutel and J. Laskar, Frequency Map and Global Dynamics in the Solar System I: Short Period Dynamics of Massless Particles, Icarus, 152 (2001), 4-28.doi: 10.1006/icar.2000.6576.
[62]	M. Sansottera, U. Locatelli and A. Giorgilli, A semi-analytic algorithm for constructing lower dimensional elliptic tori in planetary systems, Cel. Mech. & Dyn. Astr., 111 (2011), 337-361.doi: 10.1007/s10569-011-9375-x.
[63]	L. Stefanelli, Periodic and Quasi-periodic Motions in Nearly-integrable Dissipative Systems with Application to Celestial Mechanics, Ph.D. Thesis, University of Roma "Tor Vergata'', 2011.
[64]	L. Stefanelli and U. Locatelli, Kolmogorov's normal form for equations of motion with dissipative effects, DCDS-B, 17 (2012), 2561-2593.doi: 10.3934/dcdsb.2012.17.2561.