October  2004, 11(4): 745-755. doi: 10.3934/dcds.2004.11.745

Branches of periodic orbits for the planar restricted 3-body problem

1. 

Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Received  January 2003 Revised  December 2003 Published  September 2004

We describe a method for studying the existence and the linear stability of branches of periodic solutions for a dynamical system with a parameter. We apply the method to the planar restricted 3-body problem extending the results of [A]. More precisely, we prove the existence of some continuous branches of periodic orbits with the energy or the masses of the primaries as parameters, and provide an approximation of the orbits with rigorous bounds. We prove the linear stability or instability of the orbits.
Citation: Gianni Arioli. Branches of periodic orbits for the planar restricted 3-body problem. Discrete & Continuous Dynamical Systems - A, 2004, 11 (4) : 745-755. doi: 10.3934/dcds.2004.11.745
[1]

Qunyao Yin, Shiqing Zhang. New periodic solutions for the circular restricted 3-body and 4-body problems. Communications on Pure & Applied Analysis, 2010, 9 (1) : 249-260. doi: 10.3934/cpaa.2010.9.249

[2]

Giovanni F. Gronchi, Chiara Tardioli. The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1323-1344. doi: 10.3934/dcdsb.2013.18.1323

[3]

Elbaz I. Abouelmagd, Juan Luis García Guirao, Jaume Llibre. Periodic orbits for the perturbed planar circular restricted 3–body problem. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1007-1020. doi: 10.3934/dcdsb.2019003

[4]

Qinglong Zhou, Yongchao Zhang. Analytic results for the linear stability of the equilibrium point in Robe's restricted elliptic three-body problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1763-1787. doi: 10.3934/dcds.2017074

[5]

Hildeberto E. Cabral, Zhihong Xia. Subharmonic solutions in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 463-474. doi: 10.3934/dcds.1995.1.463

[6]

Samuel R. Kaplan, Ernesto A. Lacomba, Jaume Llibre. Symbolic dynamics of the elliptic rectilinear restricted 3--body problem. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 541-555. doi: 10.3934/dcdss.2008.1.541

[7]

Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009

[8]

Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229

[9]

Alain Chenciner, Jacques Féjoz. The flow of the equal-mass spatial 3-body problem in the neighborhood of the equilateral relative equilibrium. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 421-438. doi: 10.3934/dcdsb.2008.10.421

[10]

Niraj Pathak, V. O. Thomas, Elbaz I. Abouelmagd. The perturbed photogravitational restricted three-body problem: Analysis of resonant periodic orbits. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 849-875. doi: 10.3934/dcdss.2019057

[11]

Hadia H. Selim, Juan L. G. Guirao, Elbaz I. Abouelmagd. Libration points in the restricted three-body problem: Euler angles, existence and stability. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 703-710. doi: 10.3934/dcdss.2019044

[12]

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

[13]

Eduardo S. G. Leandro. On the Dziobek configurations of the restricted $(N+1)$-body problem with equal masses. Discrete & Continuous Dynamical Systems - S, 2008, 1 (4) : 589-595. doi: 10.3934/dcdss.2008.1.589

[14]

Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062

[15]

Regina Martínez, Carles Simó. On the stability of the Lagrangian homographic solutions in a curved three-body problem on $\mathbb{S}^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1157-1175. doi: 10.3934/dcds.2013.33.1157

[16]

Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-13. doi: 10.3934/dcdss.2020067

[17]

Shiqing Zhang, Qing Zhou. Nonplanar and noncollision periodic solutions for $N$-body problems. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 679-685. doi: 10.3934/dcds.2004.10.679

[18]

Jifeng Chu, Pedro J. Torres, Feng Wang. Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1921-1932. doi: 10.3934/dcds.2015.35.1921

[19]

Martha Alvarez-Ramírez, Joaquín Delgado. Blow up of the isosceles 3--body problem with an infinitesimal mass. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1149-1173. doi: 10.3934/dcds.2003.9.1149

[20]

Nai-Chia Chen. Symmetric periodic orbits in three sub-problems of the $N$-body problem. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1523-1548. doi: 10.3934/dcdsb.2014.19.1523

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]