-
Previous Article
Properly-degenerate KAM theory (following V. I. Arnold)
- DCDS-S Home
- This Issue
-
Next Article
A geometric fractional monodromy theorem
Some KAM applications to Celestial Mechanics
| 1. | Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma |
References:
| [1] |
V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, 18 (1963), 91-192. |
| [2] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics," Encyclopaedia of Mathematical Sciences 3, Dynamical Systems III, Third edition, Springer-Verlag, Berlin, 2006. |
| [3] |
A. Berretti, A. Celletti, L. Chierchia and C. Falcolini, Natural boundaries for area-preserving twist maps, J. Stat. Phys., 66 (1992), 1613-1630.
doi: 10.1007/BF01054437. |
| [4] |
L. Biasco, L. Chierchia and E. Valdinoci, Elliptic two-dimensional invariant tori for the planetary three-body problem, Arch. Rational Mech. Anal., 170 (2003), 91-135.
doi: 10.1007/s00205-003-0269-2. |
| [5] |
L. Biasco, L. Chierchia and E. Valdinoci, $N$-dimensional elliptic invariant tori for the planar $(N+1)$-body problem, SIAM Journal on Mathematical Analysis, 37 (2006), 1560-1588.
doi: 10.1137/S0036141004443646. |
| [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. Simó 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] |
A. Celletti, Analysis of resonances in the spin-orbit problem in celestial mechanics: The synchronous resonance (Part I), J. of Applied Math. and Physics (ZAMP), 41 (1990), 174-204. |
| [9] |
A. Celletti, Construction of librational invariant tori in the spin-orbit problem, J. of Applied Math. and Physics (ZAMP), 45 (1994), 61-80.
doi: 10.1007/BF00942847. |
| [10] |
A. Celletti and L. Chierchia, "KAM Stability and Celestial Mechanics," Memoirs American Mathematical Society, 187 (2007). |
| [11] |
A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics, Arch. Rational Mech. Anal., 191 (2009), 311-345.
doi: 10.1007/s00205-008-0141-5. |
| [12] |
J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman), Ergod. Th. Dynam. Sys., 24 (2004), 1521-1582. |
| [13] |
A. Giorgilli, U. Locatelli and M. Sansottera, Kolmogorov and Nekhoroshev theory for the problem of three bodies, Celest. Mech. Dyn. Astr., 104 (2009), 159-173.
doi: 10.1007/s10569-009-9192-7. |
| [14] |
J. M. Greene, A method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201.
doi: 10.1063/1.524170. |
| [15] |
M. Hénon, Exploration numérique du problème restreint IV: Masses égales, orbites non périodiques, Bulletin Astronomique, fasc. 2, 3 (1966), 49-66. |
| [16] |
A. Ya. Khinchin, "Continued Fractions," The University of Chicago Press, Chicago-London, 1964. |
| [17] |
A. N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian, Dokl. Akad. Nauk SSSR, 98 (1954), 527-530. |
| [18] |
W. H. Jefferys and J. Moser, Quasi-periodic solutions for the three-body problem, Astron. J., 71 (1966), 568-578.
doi: 10.1086/109964. |
| [19] |
À. Jorba and J. Villanueva, Effective stability around periodic orbits of the spatial RTBP, In: Hamiltonian Systems with Three or More Degrees of Freedom, NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci. 533, C. Simó ed., Kluwer Acad. Publ., Dordrecht, (1999), 628-632. |
| [20] |
J. Laskar and P. Robutel, Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian, Celest. Mech. Dyn. Astr., 62 (1995), 193-217.
doi: 10.1007/BF00692088. |
| [21] |
P. Le Calvez, Existence d'orbites quasi-périodiques dans les attracteurs de Birkhoff, Comm. Math. Phys., 106 (1986), 383-394.
doi: 10.1007/BF01207253. |
| [22] |
B. B. Lieberman, Existence of quasi-periodic solutions to the three-body problem, Celestial Mechanics, 3 (1971), 408-426.
doi: 10.1007/BF01227790. |
| [23] |
R. de la Llave and C. E. Wayne, Whiskered and low dimensional tori in nearly integrable Hamiltonian systems, MPEJ, 10 (2004). |
| [24] |
U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems, Celest. Mech. Dyn. Astr., 78 (2000), 47-74.
doi: 10.1023/A:1011139523256. |
| [25] |
U. Locatelli and A. Giorgilli, Construction of Kolmogorov's normal form for a planetary system, Reg. Chaotic Dyn., 10 (2005), 153-171.
doi: 10.1070/RD2005v010n02ABEH000309. |
| [26] |
J. Moser, On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. Göttingen, Math.-Phys. Kl. II, (1962), 1-20. |
| [27] |
G. Pinzari, "On the Kolmogorov set for Many-Body Problems," Ph.D. Thesis, Università Roma Tre 2009. |
| [28] |
P. Robutel, Stability of the planetary three-body problem. II. KAM theory and existence of quasi-periodic motions, Celest. Mech. Dyn. Astr., 62 (1995), 219-261.
doi: 10.1007/BF00692089. |
show all references
References:
| [1] |
V. I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, 18 (1963), 91-192. |
| [2] |
V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, "Mathematical Aspects of Classical and Celestial Mechanics," Encyclopaedia of Mathematical Sciences 3, Dynamical Systems III, Third edition, Springer-Verlag, Berlin, 2006. |
| [3] |
A. Berretti, A. Celletti, L. Chierchia and C. Falcolini, Natural boundaries for area-preserving twist maps, J. Stat. Phys., 66 (1992), 1613-1630.
doi: 10.1007/BF01054437. |
| [4] |
L. Biasco, L. Chierchia and E. Valdinoci, Elliptic two-dimensional invariant tori for the planetary three-body problem, Arch. Rational Mech. Anal., 170 (2003), 91-135.
doi: 10.1007/s00205-003-0269-2. |
| [5] |
L. Biasco, L. Chierchia and E. Valdinoci, $N$-dimensional elliptic invariant tori for the planar $(N+1)$-body problem, SIAM Journal on Mathematical Analysis, 37 (2006), 1560-1588.
doi: 10.1137/S0036141004443646. |
| [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. Simó 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] |
A. Celletti, Analysis of resonances in the spin-orbit problem in celestial mechanics: The synchronous resonance (Part I), J. of Applied Math. and Physics (ZAMP), 41 (1990), 174-204. |
| [9] |
A. Celletti, Construction of librational invariant tori in the spin-orbit problem, J. of Applied Math. and Physics (ZAMP), 45 (1994), 61-80.
doi: 10.1007/BF00942847. |
| [10] |
A. Celletti and L. Chierchia, "KAM Stability and Celestial Mechanics," Memoirs American Mathematical Society, 187 (2007). |
| [11] |
A. Celletti and L. Chierchia, Quasi-periodic attractors in celestial mechanics, Arch. Rational Mech. Anal., 191 (2009), 311-345.
doi: 10.1007/s00205-008-0141-5. |
| [12] |
J. Féjoz, Démonstration du 'théorème d'Arnold' sur la stabilité du système planétaire (d'après Herman), Ergod. Th. Dynam. Sys., 24 (2004), 1521-1582. |
| [13] |
A. Giorgilli, U. Locatelli and M. Sansottera, Kolmogorov and Nekhoroshev theory for the problem of three bodies, Celest. Mech. Dyn. Astr., 104 (2009), 159-173.
doi: 10.1007/s10569-009-9192-7. |
| [14] |
J. M. Greene, A method for determining a stochastic transition, J. Math. Phys., 20 (1979), 1183-1201.
doi: 10.1063/1.524170. |
| [15] |
M. Hénon, Exploration numérique du problème restreint IV: Masses égales, orbites non périodiques, Bulletin Astronomique, fasc. 2, 3 (1966), 49-66. |
| [16] |
A. Ya. Khinchin, "Continued Fractions," The University of Chicago Press, Chicago-London, 1964. |
| [17] |
A. N. Kolmogorov, On the conservation of conditionally periodic motions under small perturbation of the Hamiltonian, Dokl. Akad. Nauk SSSR, 98 (1954), 527-530. |
| [18] |
W. H. Jefferys and J. Moser, Quasi-periodic solutions for the three-body problem, Astron. J., 71 (1966), 568-578.
doi: 10.1086/109964. |
| [19] |
À. Jorba and J. Villanueva, Effective stability around periodic orbits of the spatial RTBP, In: Hamiltonian Systems with Three or More Degrees of Freedom, NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci. 533, C. Simó ed., Kluwer Acad. Publ., Dordrecht, (1999), 628-632. |
| [20] |
J. Laskar and P. Robutel, Stability of the planetary three-body problem. I. Expansion of the planetary Hamiltonian, Celest. Mech. Dyn. Astr., 62 (1995), 193-217.
doi: 10.1007/BF00692088. |
| [21] |
P. Le Calvez, Existence d'orbites quasi-périodiques dans les attracteurs de Birkhoff, Comm. Math. Phys., 106 (1986), 383-394.
doi: 10.1007/BF01207253. |
| [22] |
B. B. Lieberman, Existence of quasi-periodic solutions to the three-body problem, Celestial Mechanics, 3 (1971), 408-426.
doi: 10.1007/BF01227790. |
| [23] |
R. de la Llave and C. E. Wayne, Whiskered and low dimensional tori in nearly integrable Hamiltonian systems, MPEJ, 10 (2004). |
| [24] |
U. Locatelli and A. Giorgilli, Invariant tori in the secular motions of the three-body planetary systems, Celest. Mech. Dyn. Astr., 78 (2000), 47-74.
doi: 10.1023/A:1011139523256. |
| [25] |
U. Locatelli and A. Giorgilli, Construction of Kolmogorov's normal form for a planetary system, Reg. Chaotic Dyn., 10 (2005), 153-171.
doi: 10.1070/RD2005v010n02ABEH000309. |
| [26] |
J. Moser, On invariant curves of area-preserving mappings of an annulus, Nach. Akad. Wiss. Göttingen, Math.-Phys. Kl. II, (1962), 1-20. |
| [27] |
G. Pinzari, "On the Kolmogorov set for Many-Body Problems," Ph.D. Thesis, Università Roma Tre 2009. |
| [28] |
P. Robutel, Stability of the planetary three-body problem. II. KAM theory and existence of quasi-periodic motions, Celest. Mech. Dyn. Astr., 62 (1995), 219-261.
doi: 10.1007/BF00692089. |
| [1] |
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 |
| [2] |
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 |
| [3] |
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 |
| [4] |
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 |
| [5] |
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 |
| [6] |
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 |
| [7] |
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 |
| [8] |
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 |
| [9] |
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 |
| [10] |
Hans Koch. On the renormalization of Hamiltonian flows, and critical invariant tori. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 633-646. doi: 10.3934/dcds.2002.8.633 |
| [11] |
Xiaocai Wang. Non-floquet invariant tori in reversible systems. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3439-3457. doi: 10.3934/dcds.2018147 |
| [12] |
Eduard Feireisl, Šárka Nečasová, Reimund Rautmann, Werner Varnhorn. New developments in mathematical theory of fluid mechanics. Discrete and Continuous Dynamical Systems - S, 2014, 7 (5) : i-ii. doi: 10.3934/dcdss.2014.7.5i |
| [13] |
Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413 |
| [14] |
Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069 |
| [15] |
Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080 |
| [16] |
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 |
| [17] |
C. Chandre. Renormalization for cubic frequency invariant tori in Hamiltonian systems with two degrees of freedom. Discrete and Continuous Dynamical Systems - B, 2002, 2 (3) : 457-465. doi: 10.3934/dcdsb.2002.2.457 |
| [18] |
Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098 |
| [19] |
Denis G. Gaidashev. Renormalization of isoenergetically degenerate hamiltonian flows and associated bifurcations of invariant tori. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 63-102. doi: 10.3934/dcds.2005.13.63 |
| [20] |
Shengqing Hu. Persistence of invariant tori for almost periodically forced reversible systems. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4497-4518. doi: 10.3934/dcds.2020188 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]