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January  2017, 37(1): 229-256. doi: 10.3934/dcds.2017009

Oscillatory orbits in the restricted elliptic planar three body problem

Marcel Guardia 1,, , Tere M. Seara 1, , Pau Martín 2, and  Lara Sabbagh 3,

1. 

Departament de Matemàtiques, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain
2. 

Departament de Matemàtiques, Universitat Politècnica de Catalunya, Campus Nord, Edifici C3, C. Jordi Girona, 1-3, 08034 Barcelona, Spain
3. 

Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom

* Corresponding author: Marcel Guardia

Received  March 2016 Revised  September 2016 Published  November 2016

Fund Project: M.G., P. M. and T. S. are partially supported by the Spanish MINECO-FEDER Grant MTM2015-65715-P and the Catalan Grant 2014SGR504. T.S. is also supported by by the Russian Scientific Foundation grant 14-41-00044. and European Marie Curie Action FP7-PEOPLE-2012-IRSES: BREUDS L. S. is partially supported by the EPSRC grant EP/J003948/1

The restricted planar elliptic three body problem models the motion of a massless body under the Newtonian gravitational force of two other bodies, the primaries, which evolve in Keplerian ellipses.

A trajectory is called oscillatory if it leaves every bounded region but returns infinitely often to some fixed bounded region. We prove the existence of such type of trajectories for any values for the masses of the primaries provided the eccentricity of the Keplerian ellipses is small.

Keywords: Restricted three body problem, final motions, oscillatory motions, parabolic points, lambda lemma.
Mathematics Subject Classification: Primary:70F15, 37J4.
Citation: Marcel Guardia, Tere M. Seara, Pau Martín, Lara Sabbagh. Oscillatory orbits in the restricted elliptic planar three body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 229-256. doi: 10.3934/dcds.2017009
References:
[1]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Dynamical Systems Ⅲ volume 3 of Encyclopaedia Math. Sci. Springer, Berlin, 2006. Google Scholar

[2]

V.M. Alekseev, Quasirandom dynamical systems. Ⅰ, Ⅱ, Ⅲ, Math. USSR, 5, 6, 7, (), 1968-1969. Google Scholar

[3]

V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady, 5 (1964), 581-585. Google Scholar

[4]

I. Baldomá and E. Fontich, Stable manifolds associated to fixed points with linear part equal to identity, J. Differential Equations, 197 (2004), 45-72. doi: 10.1016/j.jde.2003.07.005. Google Scholar

[5]

I. Baldomá, E. Fontich, R. de la Llave and P. Martín, The parametrization method for one-dimensional invariant manifolds of higher dimensional parabolic fixed points, Discrete Cont. Dyn. S., 17 (2007), 835-865. doi: 10.3934/dcds.2007.17.835. Google Scholar

[6]

S. Bolotin, Symbolic dynamics of almost collision orbits and skew products of symplectic maps, Nonlinearity, 19 (2006), 2041-2063. doi: 10.1088/0951-7715/19/9/003. Google Scholar

[7]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. Ⅰ. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328. doi: 10.1512/iumj.2003.52.2245. Google Scholar

[8]

J. Chazy, Sur l'allure du mouvement dans le probléme des trois corps quand le temps croît indéfiniment, Annales scientifiques de l'École Normale Supérieure, 39 (1922), 29-130. Google Scholar

[9]

J. Cresson, A $ λ$-lemma for partially hyperbolic tori and the obstruction property, Lett. Math. Phys., 42 (1997), 363-377. doi: 10.1023/A:1007433819941. Google Scholar

[10]

J. Cresson, The transfer lemma for Graff tori and Arnold diffusion time, Discrete Contin. Dynam. Systems, 7 (2001), 787-800. doi: 10.3934/dcds.2001.7.787. Google Scholar

[11]

A. Delshams, R. de la Llave and T. M. Seara, A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of $ \mathbb{T}\sp 2$, Comm. Math. Phys., 209 (2000), 353-392. doi: 10.1007/PL00020961. Google Scholar

[12]

A. Delshams, R. de la Llave and T. M. Seara, A Geometric Mechanism for Diffusion in Hamiltonian Systems Overcoming the Large Gap Problem: Heuristics and Rigorous Verification on a Model, Mem. Amer. Math. Soc. , 2006. doi: 10.1090/memo/0844. Google Scholar

[13]

A. Delshams, R. de la Llave and T. M. Seara, Geometric properties of the scattering map of a normally hyperbolic invariant manifold, Adv. Math., 217 (2008), 1096-1153. doi: 10.1016/j.aim.2007.08.014. Google Scholar

[14]

A. Delshams, M. Gidea and P. Roldán, Transition map and shadowing lemma for normally hyperbolic invariant manifolds, Discrete Contin. Dyn. Syst., 33 (2013), 1089-1112. Google Scholar

[15]

A. Delshams, V. Kaloshin, A. de la Rosa and T. M. Seara, Global instability in the elliptic restricted three body problem, Preprint, available at http://arxiv.org/abs/1501.01214, 2014.Google Scholar

[16]

E. Fontich and P. Martín, Differentiable invariant manifolds for partially hyperbolic tori and a lambda lemma, Nonlinearity, 13 (2000), 1561-1593. doi: 10.1088/0951-7715/13/5/309. Google Scholar

[17]

J. Galante and V. Kaloshin, Destruction of invariant curves using the ordering condition, Preprint, available at http://www.terpconnect.umd.edu/~vkaloshi, 2010.Google Scholar

[18]

J. Galante and V. Kaloshin, The method of spreading cumulative twist and its application to the restricted circular planar three body problem, Preprint, available at http://www.terpconnect.umd.edu/~vkaloshi, 2010.Google Scholar

[19]

J. Galante and V. Kaloshin, Destruction of invariant curves in the restricted circular planar three-body problem by using comparison of action, Duke Math. J., 159 (2011), 275-327. doi: 10.1215/00127094-1415878. Google Scholar

[20]

A. Gorodetski and V. Kaloshin, Hausdorff dimension of oscillatory motions for restricted three body problems, Preprint, available at http://www.terpconnect.umd.edu/~vkaloshi, 2012.Google Scholar

[21]

M. Guardia, P. Martín and T. M. Seara, Oscillatory motions for the restricted planar circular three body problem, Inventiones mathematicae, 203 (2016), 417-492. doi: 10.1007/s00222-015-0591-y. Google Scholar

[22]

M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58. doi: 10.1016/j.jde.2004.03.013. Google Scholar

[23]

M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems. Ⅱ, J. Differential Equations, 202 (2004), 59-80. doi: 10.1016/j.jde.2004.03.014. Google Scholar

[24]

M. R. Herman, Sur les Courbes Invariantes Par Les Difféomorphismes de L'anneau. Vol. 1, volume 103 of Astérisque, Société Mathématique de France, Paris, 1983. Google Scholar

[25]

P. Le Calvez, Drift orbits for families of twist maps of the annulus, Ergodic Theory Dynam. Systems, 27 (2007), 869-879. doi: 10.1017/S0143385706000903. Google Scholar

[26]

J. Llibre and C. Simó, Oscillatory solutions in the planar restricted three-body problem, Math. Ann., 248 (1980), 153-184. doi: 10.1007/BF01421955. Google Scholar

[27]

J. Llibre and C. Simó, Some homoclinic phenomena in the three-body problem, J. Differential Equations, 37 (1980), 444-465. doi: 10.1016/0022-0396(80)90109-6. Google Scholar

[28]

J. P. Marco, Transition le long des chaî nes de tores invariants pour les systèmes hamiltoniens analytiques, Ann. Inst. H. Poincaré Phys. Théor., 64 (1996), 205-252. Google Scholar

[29]

R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Differential Equations, 14 (1973), 70-88. doi: 10.1016/0022-0396(73)90077-6. Google Scholar

[30]

K. R. Meyer and G. R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4757-4073-8. Google Scholar

[31]

R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem, SIAM Journal of Mathematical Analysis, 15 (1984), 857-876. doi: 10.1137/0515065. Google Scholar

[32]

R. Moeckel, Symbolic dynamics in the planar three-body problem, Regul. Chaotic Dyn., 12 (2007), 449-475. doi: 10.1134/S1560354707050012. Google Scholar

[33]

J. Moser, Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics, Princeton University Press, Princeton, N. J. , 1973. Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J, Annals of Mathematics Studies, No. 77. Google Scholar

[34]

C. Robinson, Homoclinic orbits and oscillation for the planar three-body problem, J. Differential Equations, 52 (1984), 356-377. doi: 10.1016/0022-0396(84)90168-2. Google Scholar

[35]

C. Robinson, Topological decoupling near planar parabolic orbits, Qualitative Theory of Dynamical Systems, 14 (2015), 337-351. doi: 10.1007/s12346-015-0130-7. Google Scholar

[36]

L. Sabbagh, An inclination lemma for normally hyperbolic manifolds with an application to diffusion, Ergodic Theory Dynam. Systems, 35 (2015), 2269-2291. doi: 10.1017/etds.2014.30. Google Scholar

[37]

L. P. Šil'nikov, On a problem of Poincaré-Birkhoff, Mat. Sb. (N.S.), 74 (1967), 378-397. Google Scholar

[38]

K. Sitnikov, The existence of oscillatory motions in the three-body problems, Soviet Physics. Dokl., 5 (1960), 647-650. Google Scholar

[39]

Z. Xia, Mel'cprime nikov method and transversal homoclinic points in the restricted three-body problem, J. Differential Equations, 96 (1992), 170-184. doi: 10.1016/0022-0396(92)90149-H. Google Scholar

[40]

Z. Xia, Arnol$'$d diffusion and oscillatory solutions in the planar three-body problem, J. Differential Equations, 110 (1994), 289-321. doi: 10.1006/jdeq.1994.1069. Google Scholar

show all references

References:
[1]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Dynamical Systems Ⅲ volume 3 of Encyclopaedia Math. Sci. Springer, Berlin, 2006. Google Scholar

[2]

V.M. Alekseev, Quasirandom dynamical systems. Ⅰ, Ⅱ, Ⅲ, Math. USSR, 5, 6, 7, (), 1968-1969. Google Scholar

[3]

V. I. Arnold, Instability of dynamical systems with several degrees of freedom, Sov. Math. Doklady, 5 (1964), 581-585. Google Scholar

[4]

I. Baldomá and E. Fontich, Stable manifolds associated to fixed points with linear part equal to identity, J. Differential Equations, 197 (2004), 45-72. doi: 10.1016/j.jde.2003.07.005. Google Scholar

[5]

I. Baldomá, E. Fontich, R. de la Llave and P. Martín, The parametrization method for one-dimensional invariant manifolds of higher dimensional parabolic fixed points, Discrete Cont. Dyn. S., 17 (2007), 835-865. doi: 10.3934/dcds.2007.17.835. Google Scholar

[6]

S. Bolotin, Symbolic dynamics of almost collision orbits and skew products of symplectic maps, Nonlinearity, 19 (2006), 2041-2063. doi: 10.1088/0951-7715/19/9/003. Google Scholar

[7]

X. Cabré, E. Fontich and R. de la Llave, The parameterization method for invariant manifolds. Ⅰ. Manifolds associated to non-resonant subspaces, Indiana Univ. Math. J., 52 (2003), 283-328. doi: 10.1512/iumj.2003.52.2245. Google Scholar

[8]

J. Chazy, Sur l'allure du mouvement dans le probléme des trois corps quand le temps croît indéfiniment, Annales scientifiques de l'École Normale Supérieure, 39 (1922), 29-130. Google Scholar

[9]

J. Cresson, A $ λ$-lemma for partially hyperbolic tori and the obstruction property, Lett. Math. Phys., 42 (1997), 363-377. doi: 10.1023/A:1007433819941. Google Scholar

[10]

J. Cresson, The transfer lemma for Graff tori and Arnold diffusion time, Discrete Contin. Dynam. Systems, 7 (2001), 787-800. doi: 10.3934/dcds.2001.7.787. Google Scholar

[11]

A. Delshams, R. de la Llave and T. M. Seara, A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of $ \mathbb{T}\sp 2$, Comm. Math. Phys., 209 (2000), 353-392. doi: 10.1007/PL00020961. Google Scholar

[12]

A. Delshams, R. de la Llave and T. M. Seara, A Geometric Mechanism for Diffusion in Hamiltonian Systems Overcoming the Large Gap Problem: Heuristics and Rigorous Verification on a Model, Mem. Amer. Math. Soc. , 2006. doi: 10.1090/memo/0844. Google Scholar

[13]

A. Delshams, R. de la Llave and T. M. Seara, Geometric properties of the scattering map of a normally hyperbolic invariant manifold, Adv. Math., 217 (2008), 1096-1153. doi: 10.1016/j.aim.2007.08.014. Google Scholar

[14]

A. Delshams, M. Gidea and P. Roldán, Transition map and shadowing lemma for normally hyperbolic invariant manifolds, Discrete Contin. Dyn. Syst., 33 (2013), 1089-1112. Google Scholar

[15]

A. Delshams, V. Kaloshin, A. de la Rosa and T. M. Seara, Global instability in the elliptic restricted three body problem, Preprint, available at http://arxiv.org/abs/1501.01214, 2014.Google Scholar

[16]

E. Fontich and P. Martín, Differentiable invariant manifolds for partially hyperbolic tori and a lambda lemma, Nonlinearity, 13 (2000), 1561-1593. doi: 10.1088/0951-7715/13/5/309. Google Scholar

[17]

J. Galante and V. Kaloshin, Destruction of invariant curves using the ordering condition, Preprint, available at http://www.terpconnect.umd.edu/~vkaloshi, 2010.Google Scholar

[18]

J. Galante and V. Kaloshin, The method of spreading cumulative twist and its application to the restricted circular planar three body problem, Preprint, available at http://www.terpconnect.umd.edu/~vkaloshi, 2010.Google Scholar

[19]

J. Galante and V. Kaloshin, Destruction of invariant curves in the restricted circular planar three-body problem by using comparison of action, Duke Math. J., 159 (2011), 275-327. doi: 10.1215/00127094-1415878. Google Scholar

[20]

A. Gorodetski and V. Kaloshin, Hausdorff dimension of oscillatory motions for restricted three body problems, Preprint, available at http://www.terpconnect.umd.edu/~vkaloshi, 2012.Google Scholar

[21]

M. Guardia, P. Martín and T. M. Seara, Oscillatory motions for the restricted planar circular three body problem, Inventiones mathematicae, 203 (2016), 417-492. doi: 10.1007/s00222-015-0591-y. Google Scholar

[22]

M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems, J. Differential Equations, 202 (2004), 32-58. doi: 10.1016/j.jde.2004.03.013. Google Scholar

[23]

M. Gidea and P. Zgliczyński, Covering relations for multidimensional dynamical systems. Ⅱ, J. Differential Equations, 202 (2004), 59-80. doi: 10.1016/j.jde.2004.03.014. Google Scholar

[24]

M. R. Herman, Sur les Courbes Invariantes Par Les Difféomorphismes de L'anneau. Vol. 1, volume 103 of Astérisque, Société Mathématique de France, Paris, 1983. Google Scholar

[25]

P. Le Calvez, Drift orbits for families of twist maps of the annulus, Ergodic Theory Dynam. Systems, 27 (2007), 869-879. doi: 10.1017/S0143385706000903. Google Scholar

[26]

J. Llibre and C. Simó, Oscillatory solutions in the planar restricted three-body problem, Math. Ann., 248 (1980), 153-184. doi: 10.1007/BF01421955. Google Scholar

[27]

J. Llibre and C. Simó, Some homoclinic phenomena in the three-body problem, J. Differential Equations, 37 (1980), 444-465. doi: 10.1016/0022-0396(80)90109-6. Google Scholar

[28]

J. P. Marco, Transition le long des chaî nes de tores invariants pour les systèmes hamiltoniens analytiques, Ann. Inst. H. Poincaré Phys. Théor., 64 (1996), 205-252. Google Scholar

[29]

R. McGehee, A stable manifold theorem for degenerate fixed points with applications to celestial mechanics, J. Differential Equations, 14 (1973), 70-88. doi: 10.1016/0022-0396(73)90077-6. Google Scholar

[30]

K. R. Meyer and G. R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-body Problem, Springer-Verlag, New York, 1992. doi: 10.1007/978-1-4757-4073-8. Google Scholar

[31]

R. Moeckel, Heteroclinic phenomena in the isosceles three-body problem, SIAM Journal of Mathematical Analysis, 15 (1984), 857-876. doi: 10.1137/0515065. Google Scholar

[32]

R. Moeckel, Symbolic dynamics in the planar three-body problem, Regul. Chaotic Dyn., 12 (2007), 449-475. doi: 10.1134/S1560354707050012. Google Scholar

[33]

J. Moser, Stable and Random Motions in Dynamical Systems, with Special Emphasis on Celestial Mechanics, Princeton University Press, Princeton, N. J. , 1973. Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J, Annals of Mathematics Studies, No. 77. Google Scholar

[34]

C. Robinson, Homoclinic orbits and oscillation for the planar three-body problem, J. Differential Equations, 52 (1984), 356-377. doi: 10.1016/0022-0396(84)90168-2. Google Scholar

[35]

C. Robinson, Topological decoupling near planar parabolic orbits, Qualitative Theory of Dynamical Systems, 14 (2015), 337-351. doi: 10.1007/s12346-015-0130-7. Google Scholar

[36]

L. Sabbagh, An inclination lemma for normally hyperbolic manifolds with an application to diffusion, Ergodic Theory Dynam. Systems, 35 (2015), 2269-2291. doi: 10.1017/etds.2014.30. Google Scholar

[37]

L. P. Šil'nikov, On a problem of Poincaré-Birkhoff, Mat. Sb. (N.S.), 74 (1967), 378-397. Google Scholar

[38]

K. Sitnikov, The existence of oscillatory motions in the three-body problems, Soviet Physics. Dokl., 5 (1960), 647-650. Google Scholar

[39]

Z. Xia, Mel'cprime nikov method and transversal homoclinic points in the restricted three-body problem, J. Differential Equations, 96 (1992), 170-184. doi: 10.1016/0022-0396(92)90149-H. Google Scholar

[40]

Z. Xia, Arnol$'$d diffusion and oscillatory solutions in the planar three-body problem, J. Differential Equations, 110 (1994), 289-321. doi: 10.1006/jdeq.1994.1069. Google Scholar

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