-
Previous Article
Existence of solutions to chemotaxis dynamics with logistic source
- PROC Home
- This Issue
-
Next Article
Blow-up of solutions to semilinear wave equations with non-zero initial data
Classification of periodic orbits in the planar equal-mass four-body problem
1. | School of Mathematics and System Science, Beihang University, Beijing 100191, China |
2. | Department of Mathematics, Brigham Young University, Provo, Utah 84602 |
3. | Department of Mathematics and Computer Science, Virginia State University, Petersburg, Virginia 23806 |
References:
[1] |
R. Broucke, Classification of periodic orbits in the four- and five-body problems, Ann. N.Y. Acad. Sci., 1017 (2004), 408-421. |
[2] |
K. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal., 170 (2001), 293-318. |
[3] |
K. Chen, Variational methods on periodic and quasi-periodic solutions for the N-body problem, Erg. Thy. Dyn. Sys., 23 (2003), 1691-1715. |
[4] |
L. Sbano, Periodic orbits of Hamiltonian systems, in Mathematics of Complexity and Dynamical Systems(ed. R.A. Meyers), Springer, (2011), 1212-1236. |
[5] |
T. Ouyang, and Z. Xie, A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4-body problem, preprint,, , ().
|
[6] |
T. Ouyang, and Z. Xie, A continuum of periodic solutions to the four-body problem with various choices of masses, preprint,, , ().
|
[7] |
D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem, Invent. Math., 155 (2004), 305-362. |
[8] |
M. Šuvakov and V. Dmitrašinović, Three classes of Newtonian three-body planar periodic orbits, Phy. Rev. Lett., 110 (2013), 114301. |
[9] |
R. Vanderbei, New orbits for the n-body problem, Ann. N.Y. Acad. Sci., 1017 (2004), 422-433. |
[10] |
L. Bakker, T. Ouyang, D. Yan, S. Simmons and G. Roberts, Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem, Celestial Mech. Dynam. Astronom., 108 (2010), 147-164. |
[11] |
D. Yan, Existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four-body problem, J. Math. Anal. Appl. 388 (2012), 942-951. |
[12] |
T. Ouyang, S. Simmons and D.Yan, Periodic solutions with singularities in two dimensions in the n-body problem, Rocky Mountain J. Math., 42 (2012), 1601-1614. |
[13] |
D. Yan, and T. Ouyang, New phenomena in the spatial isosceles three-body problem, Inter. J. Bifurcation Chaos, 25 (2015), 1550116. |
[14] |
D. Yan, and T. Ouyang, Existence and linear stability of spatial isosceles periodic orbits in the equal-mass three-body problem,, preprint., ().
|
show all references
References:
[1] |
R. Broucke, Classification of periodic orbits in the four- and five-body problems, Ann. N.Y. Acad. Sci., 1017 (2004), 408-421. |
[2] |
K. Chen, Action-minimizing orbits in the parallelogram four-body problem with equal masses, Arch. Ration. Mech. Anal., 170 (2001), 293-318. |
[3] |
K. Chen, Variational methods on periodic and quasi-periodic solutions for the N-body problem, Erg. Thy. Dyn. Sys., 23 (2003), 1691-1715. |
[4] |
L. Sbano, Periodic orbits of Hamiltonian systems, in Mathematics of Complexity and Dynamical Systems(ed. R.A. Meyers), Springer, (2011), 1212-1236. |
[5] |
T. Ouyang, and Z. Xie, A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4-body problem, preprint,, , ().
|
[6] |
T. Ouyang, and Z. Xie, A continuum of periodic solutions to the four-body problem with various choices of masses, preprint,, , ().
|
[7] |
D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical n-body problem, Invent. Math., 155 (2004), 305-362. |
[8] |
M. Šuvakov and V. Dmitrašinović, Three classes of Newtonian three-body planar periodic orbits, Phy. Rev. Lett., 110 (2013), 114301. |
[9] |
R. Vanderbei, New orbits for the n-body problem, Ann. N.Y. Acad. Sci., 1017 (2004), 422-433. |
[10] |
L. Bakker, T. Ouyang, D. Yan, S. Simmons and G. Roberts, Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem, Celestial Mech. Dynam. Astronom., 108 (2010), 147-164. |
[11] |
D. Yan, Existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four-body problem, J. Math. Anal. Appl. 388 (2012), 942-951. |
[12] |
T. Ouyang, S. Simmons and D.Yan, Periodic solutions with singularities in two dimensions in the n-body problem, Rocky Mountain J. Math., 42 (2012), 1601-1614. |
[13] |
D. Yan, and T. Ouyang, New phenomena in the spatial isosceles three-body problem, Inter. J. Bifurcation Chaos, 25 (2015), 1550116. |
[14] |
D. Yan, and T. Ouyang, Existence and linear stability of spatial isosceles periodic orbits in the equal-mass three-body problem,, preprint., ().
|
[1] |
Davide L. Ferrario, Alessandro Portaluri. Dynamics of the the dihedral four-body problem. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 925-974. doi: 10.3934/dcdss.2013.6.925 |
[2] |
Hsin-Yuan Huang. Schubart-like orbits in the Newtonian collinear four-body problem: A variational proof. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1763-1774. doi: 10.3934/dcds.2012.32.1763 |
[3] |
Tiancheng Ouyang, Zhifu Xie. Regularization of simultaneous binary collisions and solutions with singularity in the collinear four-body problem. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 909-932. doi: 10.3934/dcds.2009.24.909 |
[4] |
Anete S. Cavalcanti. An existence proof of a symmetric periodic orbit in the octahedral six-body problem. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 1903-1922. doi: 10.3934/dcds.2017080 |
[5] |
Frederic Gabern, Àngel Jorba. A restricted four-body model for the dynamics near the Lagrangian points of the Sun-Jupiter system. Discrete and Continuous Dynamical Systems - B, 2001, 1 (2) : 143-182. doi: 10.3934/dcdsb.2001.1.143 |
[6] |
Sergey V. Bolotin, Piero Negrini. Variational approach to second species periodic solutions of Poincaré of the 3 body problem. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1009-1032. doi: 10.3934/dcds.2013.33.1009 |
[7] |
Mark Lewis, Daniel Offin, Pietro-Luciano Buono, Mitchell Kovacic. Instability of the periodic hip-hop orbit in the $2N$-body problem with equal masses. Discrete and Continuous Dynamical Systems, 2013, 33 (3) : 1137-1155. doi: 10.3934/dcds.2013.33.1137 |
[8] |
Ernesto A. Lacomba, Mario Medina. Oscillatory motions in the rectangular four body problem. Discrete and Continuous Dynamical Systems - S, 2008, 1 (4) : 557-587. doi: 10.3934/dcdss.2008.1.557 |
[9] |
Hiroshi Ozaki, Hiroshi Fukuda, Toshiaki Fujiwara. Determination of motion from orbit in the three-body problem. Conference Publications, 2011, 2011 (Special) : 1158-1166. doi: 10.3934/proc.2011.2011.1158 |
[10] |
Kuo-Chang Chen. On Chenciner-Montgomery's orbit in the three-body problem. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 85-90. doi: 10.3934/dcds.2001.7.85 |
[11] |
Xiaojun Chang, Tiancheng Ouyang, Duokui Yan. Linear stability of the criss-cross orbit in the equal-mass three-body problem. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 5971-5991. doi: 10.3934/dcds.2016062 |
[12] |
Giovanni F. Gronchi, Chiara Tardioli. The evolution of the orbit distance in the double averaged restricted 3-body problem with crossing singularities. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1323-1344. doi: 10.3934/dcdsb.2013.18.1323 |
[13] |
Samuel R. Kaplan, Mark Levi, Richard Montgomery. Making the moon reverse its orbit, or, stuttering in the planar three-body problem. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 569-595. doi: 10.3934/dcdsb.2008.10.569 |
[14] |
Nai-Chia Chen. Symmetric periodic orbits in three sub-problems of the $N$-body problem. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1523-1548. doi: 10.3934/dcdsb.2014.19.1523 |
[15] |
Gianni Arioli. Branches of periodic orbits for the planar restricted 3-body problem. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 745-755. doi: 10.3934/dcds.2004.11.745 |
[16] |
Elbaz I. Abouelmagd, Juan Luis García Guirao, Jaume Llibre. Periodic orbits for the perturbed planar circular restricted 3–body problem. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1007-1020. doi: 10.3934/dcdsb.2019003 |
[17] |
Jungsoo Kang. Some remarks on symmetric periodic orbits in the restricted three-body problem. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5229-5245. doi: 10.3934/dcds.2014.34.5229 |
[18] |
Daniel Offin, Hildeberto Cabral. Hyperbolicity for symmetric periodic orbits in the isosceles three body problem. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 379-392. doi: 10.3934/dcdss.2009.2.379 |
[19] |
Tiancheng Ouyang, Duokui Yan. Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 3989-4018. doi: 10.3934/dcds.2017169 |
[20] |
Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184 |
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]