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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
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). doi: 10.1145/2382585.2382590. Google Scholar

[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. Google Scholar

[3]

D. Bambusi and E. Haus, Asymptotic stability of synchronous orbits for a gravitating viscoelastic sphere,, Cel. Mech. & Dyn. Astr., 114 (2012), 255. doi: 10.1007/s10569-012-9438-7. Google Scholar

[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. doi: 10.1007/BF02748972. Google Scholar

[5]

L. Biasco and L. Chierchia, Low-order resonances in weakly dissipative spin-orbit models,, J. Diff. Equations, 246 (2009), 4345. doi: 10.1016/j.jde.2008.11.008. Google Scholar

[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 (1996). Google Scholar

[7]

H. W. Broer, C. Simò and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms,, Nonlinearity, 11 (1998), 667. doi: 10.1088/0951-7715/11/3/015. Google Scholar

[8]

R. Calleja and A. Celletti, Breakdown of invariant attractors for the dissipative standard map,, CHAOS, 20 (2010). doi: 10.1063/1.3335408. Google Scholar

[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. doi: 10.1016/j.jde.2013.05.001. Google Scholar

[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. doi: 10.1007/s10884-013-9319-0. Google Scholar

[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. doi: 10.1088/0951-7715/23/9/001. Google Scholar

[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. doi: 10.1007/BF00945107. Google Scholar

[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. doi: 10.1007/BF00945951. Google Scholar

[14]

A. Celletti, Periodic and quasi-periodic attractors of weakly-dissipative nearly-integrable systems,, Reg. & Ch. Dyn., 14 (2009), 49. doi: 10.1134/S1560354709010067. Google Scholar

[15]

A. Celletti, Stability and Chaos in Celestial Mechanics,, Springer-Praxis, (2010). doi: 10.1007/978-3-540-85146-2. Google Scholar

[16]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics,, Memoirs American Mathematical Society, 187 (2007). doi: 10.1090/memo/0878. Google Scholar

[17]

A. Celletti and L. Chierchia, Measures of basins of attraction in spin-orbit dynamics,, Cel. Mech. & Dyn. Astr., 101 (2008), 159. doi: 10.1007/s10569-008-9142-9. Google Scholar

[18]

A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics,, Arch. Rat. Mech. Anal., 191 (2009), 311. doi: 10.1007/s00205-008-0141-5. Google Scholar

[19]

A. Celletti and S. Di Ruzza, Periodic and quasi-periodic orbits of the dissipative standard map,, DCDS-B, 16 (2011), 151. doi: 10.3934/dcdsb.2011.16.151. Google Scholar

[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. doi: 10.1140/epjst/e2010-01259-2. Google Scholar

[21]

A. Celletti, C. Froeschlé and E. Lega, Dissipative and weakly-dissipative regimes in nearly-integrable mappings,, DCDS-A, 16 (2006), 757. doi: 10.3934/dcds.2006.16.757. Google Scholar

[22]

A. Celletti, A. Giorgilli and U. Locatelli, Improved estimates on the existence of invariant tori for Hamiltonian systems,, Nonlinearity, 13 (2000), 397. doi: 10.1088/0951-7715/13/2/304. Google Scholar

[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. doi: 10.1016/S0167-2789(01)00268-8. Google Scholar

[24]

L. Chierchia, A. N. Kolmogorov's 1954 paper on nearly-integrable Hamiltonian systems,, Reg. & Ch. Dyn., 13 (2008), 130. doi: 10.1134/S1560354708020056. Google Scholar

[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. doi: 10.1038/nature02609. Google Scholar

[26]

A. Deprit and A. Deprit-Bartholomé, Stability of the triangular Lagrangian points,, Astron. J., 72 (1967). doi: 10.1086/110213. Google Scholar

[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. Google Scholar

[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. doi: 10.1103/PhysRevLett.70.2975. Google Scholar

[29]

F. Gabern, A. Jorba and U. Locatelli, On the construction of the Kolmogorov normal form for the Trojan asteroids,, Nonlinearity, 18 (2005), 1705. doi: 10.1088/0951-7715/18/4/017. Google Scholar

[30]

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations,, Prentice-Hall, (1971). Google Scholar

[31]

A. Giorgilli and U. Locatelli, Kolmogorov theorem and classical perturbation theory,, J. of App. Math. and Phys. (ZAMP), 48 (1997), 220. doi: 10.1007/PL00001475. Google Scholar

[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. doi: 10.1007/s10569-009-9192-7. Google Scholar

[33]

A. Giorgilli and M. Sansottera, Methods of algebraic manipulation in perturbation theory,, in Chaos, (2012). Google Scholar

[34]

P. Goldreich and S. J. Peale, Spin-orbit coupling in the Solar System,, Astron. J., 71 (1966), 425. doi: 10.1086/109947. Google Scholar

[35]

P. Goldreich and S. J. Peale, The dynamics of planetary rotations,, Ann. Rev. Astron. Astrophys., 6 (1968), 287. doi: 10.1146/annurev.aa.06.090168.001443. Google Scholar

[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. doi: 10.3934/dcdsb.2010.14.41. Google Scholar

[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. doi: 10.3934/dcdsb.2010.14.75. Google Scholar

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M. Govin, C. Chandre and H. R. Jauslin, KAM-Renormalization-Group analysis of stability in Hamiltonian flows,, Phys. Rev. Lett., 79 (1997), 3881. Google Scholar

[39]

J. M. Greene, A method for determining a stochastic transition,, J. of Math. Phys, 20 (1979), 1183. doi: 10.1063/1.524170. Google Scholar

[40]

E. Haus and D. Bambusi, Asymptotic behavior of an elastic satellite with internal friction,, Celestial Mechanics and Dynamical Astronomy, 114 (2012), 255. doi: 10.1007/s10569-012-9438-7. Google Scholar

[41]

M. Hénon, Exploration numérique du problème restreint IV: Masses égales, orbites non périodiques,, Bulletin Astronomique, 3 (1966), 49. Google Scholar

[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. doi: 10.1080/10586458.2005.10128904. Google Scholar

[43]

A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function,, Dokl. Akad. Nauk SSSR, 98 (1954), 527. Google Scholar

[44]

J. Laskar, Introduction to frequency map analysis,, in Hamiltonian Systems with Three or More Degrees of Freedom (ed. C. Simò), 533 (1999), 19. Google Scholar

[45]

J. Laskar, Frequency Map analysis and quasi periodic decompositions,, in Hamiltonian systems and Fourier analysis (eds. Benest et al.), (2005). Google Scholar

[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. doi: 10.1016/0167-2789(92)90028-L. Google Scholar

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J. Laskar and P. Robutel, The chaotic obliquity of the planets,, Nature, 361 (1993), 608. doi: 10.1038/361608a0. Google Scholar

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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. doi: 10.1016/0167-2789(96)00046-2. Google Scholar

[49]

A. M. Leontovich, On the stability of the Lagrange periodic solutions for the reduced problem of three bodies,, Soviet Math. Dokl., 3 (1962). Google Scholar

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

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U. Locatelli and A. Giorgilli, Construction of the Kolmogorov's normal form for a planetary system,, Reg. & Ch. Dyn., 10 (2005), 153. doi: 10.1070/RD2005v010n02ABEH000309. Google Scholar

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U. Locatelli and A. Giorgilli, Invariant tori in the Sun-Jupiter-Saturn system,, DCDS-B, 7 (2007), 377. doi: 10.3934/dcdsb.2007.7.377. Google Scholar

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G. J. F. MacDonald, Tidal friction,, Rev. Geophys., 2 (1964), 467. doi: 10.1029/RG002i003p00467. Google Scholar

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R. S. MacKay, Greene's residue criterion,, Nonlinearity, 5 (1992), 161. doi: 10.1088/0951-7715/5/1/007. Google Scholar

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A. Morbidelli and A. Giorgilli, Superexponential stability of KAM tori,, J. Stat. Phys., 78 (1995), 1607. doi: 10.1007/BF02180145. Google Scholar

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L. Stefanelli and U. Locatelli, Kolmogorov's normal form for equations of motion with dissipative effects,, DCDS-B, 17 (2012), 2561. doi: 10.3934/dcdsb.2012.17.2561. Google Scholar

show all references

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). doi: 10.1145/2382585.2382590. Google Scholar

[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. Google Scholar

[3]

D. Bambusi and E. Haus, Asymptotic stability of synchronous orbits for a gravitating viscoelastic sphere,, Cel. Mech. & Dyn. Astr., 114 (2012), 255. doi: 10.1007/s10569-012-9438-7. Google Scholar

[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. doi: 10.1007/BF02748972. Google Scholar

[5]

L. Biasco and L. Chierchia, Low-order resonances in weakly dissipative spin-orbit models,, J. Diff. Equations, 246 (2009), 4345. doi: 10.1016/j.jde.2008.11.008. Google Scholar

[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 (1996). Google Scholar

[7]

H. W. Broer, C. Simò and J. C. Tatjer, Towards global models near homoclinic tangencies of dissipative diffeomorphisms,, Nonlinearity, 11 (1998), 667. doi: 10.1088/0951-7715/11/3/015. Google Scholar

[8]

R. Calleja and A. Celletti, Breakdown of invariant attractors for the dissipative standard map,, CHAOS, 20 (2010). doi: 10.1063/1.3335408. Google Scholar

[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. doi: 10.1016/j.jde.2013.05.001. Google Scholar

[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. doi: 10.1007/s10884-013-9319-0. Google Scholar

[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. doi: 10.1088/0951-7715/23/9/001. Google Scholar

[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. doi: 10.1007/BF00945107. Google Scholar

[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. doi: 10.1007/BF00945951. Google Scholar

[14]

A. Celletti, Periodic and quasi-periodic attractors of weakly-dissipative nearly-integrable systems,, Reg. & Ch. Dyn., 14 (2009), 49. doi: 10.1134/S1560354709010067. Google Scholar

[15]

A. Celletti, Stability and Chaos in Celestial Mechanics,, Springer-Praxis, (2010). doi: 10.1007/978-3-540-85146-2. Google Scholar

[16]

A. Celletti and L. Chierchia, KAM stability and celestial mechanics,, Memoirs American Mathematical Society, 187 (2007). doi: 10.1090/memo/0878. Google Scholar

[17]

A. Celletti and L. Chierchia, Measures of basins of attraction in spin-orbit dynamics,, Cel. Mech. & Dyn. Astr., 101 (2008), 159. doi: 10.1007/s10569-008-9142-9. Google Scholar

[18]

A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics,, Arch. Rat. Mech. Anal., 191 (2009), 311. doi: 10.1007/s00205-008-0141-5. Google Scholar

[19]

A. Celletti and S. Di Ruzza, Periodic and quasi-periodic orbits of the dissipative standard map,, DCDS-B, 16 (2011), 151. doi: 10.3934/dcdsb.2011.16.151. Google Scholar

[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. doi: 10.1140/epjst/e2010-01259-2. Google Scholar

[21]

A. Celletti, C. Froeschlé and E. Lega, Dissipative and weakly-dissipative regimes in nearly-integrable mappings,, DCDS-A, 16 (2006), 757. doi: 10.3934/dcds.2006.16.757. Google Scholar

[22]

A. Celletti, A. Giorgilli and U. Locatelli, Improved estimates on the existence of invariant tori for Hamiltonian systems,, Nonlinearity, 13 (2000), 397. doi: 10.1088/0951-7715/13/2/304. Google Scholar

[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. doi: 10.1016/S0167-2789(01)00268-8. Google Scholar

[24]

L. Chierchia, A. N. Kolmogorov's 1954 paper on nearly-integrable Hamiltonian systems,, Reg. & Ch. Dyn., 13 (2008), 130. doi: 10.1134/S1560354708020056. Google Scholar

[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. doi: 10.1038/nature02609. Google Scholar

[26]

A. Deprit and A. Deprit-Bartholomé, Stability of the triangular Lagrangian points,, Astron. J., 72 (1967). doi: 10.1086/110213. Google Scholar

[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. Google Scholar

[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. doi: 10.1103/PhysRevLett.70.2975. Google Scholar

[29]

F. Gabern, A. Jorba and U. Locatelli, On the construction of the Kolmogorov normal form for the Trojan asteroids,, Nonlinearity, 18 (2005), 1705. doi: 10.1088/0951-7715/18/4/017. Google Scholar

[30]

C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations,, Prentice-Hall, (1971). Google Scholar

[31]

A. Giorgilli and U. Locatelli, Kolmogorov theorem and classical perturbation theory,, J. of App. Math. and Phys. (ZAMP), 48 (1997), 220. doi: 10.1007/PL00001475. Google Scholar

[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. doi: 10.1007/s10569-009-9192-7. Google Scholar

[33]

A. Giorgilli and M. Sansottera, Methods of algebraic manipulation in perturbation theory,, in Chaos, (2012). Google Scholar

[34]

P. Goldreich and S. J. Peale, Spin-orbit coupling in the Solar System,, Astron. J., 71 (1966), 425. doi: 10.1086/109947. Google Scholar

[35]

P. Goldreich and S. J. Peale, The dynamics of planetary rotations,, Ann. Rev. Astron. Astrophys., 6 (1968), 287. doi: 10.1146/annurev.aa.06.090168.001443. Google Scholar

[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. doi: 10.3934/dcdsb.2010.14.41. Google Scholar

[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. doi: 10.3934/dcdsb.2010.14.75. Google Scholar

[38]

M. Govin, C. Chandre and H. R. Jauslin, KAM-Renormalization-Group analysis of stability in Hamiltonian flows,, Phys. Rev. Lett., 79 (1997), 3881. Google Scholar

[39]

J. M. Greene, A method for determining a stochastic transition,, J. of Math. Phys, 20 (1979), 1183. doi: 10.1063/1.524170. Google Scholar

[40]

E. Haus and D. Bambusi, Asymptotic behavior of an elastic satellite with internal friction,, Celestial Mechanics and Dynamical Astronomy, 114 (2012), 255. doi: 10.1007/s10569-012-9438-7. Google Scholar

[41]

M. Hénon, Exploration numérique du problème restreint IV: Masses égales, orbites non périodiques,, Bulletin Astronomique, 3 (1966), 49. Google Scholar

[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. doi: 10.1080/10586458.2005.10128904. Google Scholar

[43]

A. N. Kolmogorov, Preservation of conditionally periodic movements with small change in the Hamilton function,, Dokl. Akad. Nauk SSSR, 98 (1954), 527. Google Scholar

[44]

J. Laskar, Introduction to frequency map analysis,, in Hamiltonian Systems with Three or More Degrees of Freedom (ed. C. Simò), 533 (1999), 19. Google Scholar

[45]

J. Laskar, Frequency Map analysis and quasi periodic decompositions,, in Hamiltonian systems and Fourier analysis (eds. Benest et al.), (2005). Google Scholar

[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. doi: 10.1016/0167-2789(92)90028-L. Google Scholar

[47]

J. Laskar and P. Robutel, The chaotic obliquity of the planets,, Nature, 361 (1993), 608. doi: 10.1038/361608a0. Google Scholar

[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. doi: 10.1016/0167-2789(96)00046-2. Google Scholar

[49]

A. M. Leontovich, On the stability of the Lagrange periodic solutions for the reduced problem of three bodies,, Soviet Math. Dokl., 3 (1962). Google Scholar

[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. doi: 10.1023/A:1011139523256. Google Scholar

[51]

U. Locatelli and A. Giorgilli, Construction of the Kolmogorov's normal form for a planetary system,, Reg. & Ch. Dyn., 10 (2005), 153. doi: 10.1070/RD2005v010n02ABEH000309. Google Scholar

[52]

U. Locatelli and A. Giorgilli, Invariant tori in the Sun-Jupiter-Saturn system,, DCDS-B, 7 (2007), 377. doi: 10.3934/dcdsb.2007.7.377. Google Scholar

[53]

G. J. F. MacDonald, Tidal friction,, Rev. Geophys., 2 (1964), 467. doi: 10.1029/RG002i003p00467. Google Scholar

[54]

R. S. MacKay, Greene's residue criterion,, Nonlinearity, 5 (1992), 161. doi: 10.1088/0951-7715/5/1/007. Google Scholar

[55]

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