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Classification of periodic orbits in the planar equalmass fourbody 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 fivebody problems, Ann. N.Y. Acad. Sci., 1017 (2004), 408421. 
[2] 
K. Chen, Actionminimizing orbits in the parallelogram fourbody problem with equal masses, Arch. Ration. Mech. Anal., 170 (2001), 293318. 
[3] 
K. Chen, Variational methods on periodic and quasiperiodic solutions for the Nbody problem, Erg. Thy. Dyn. Sys., 23 (2003), 16911715. 
[4] 
L. Sbano, Periodic orbits of Hamiltonian systems, in Mathematics of Complexity and Dynamical Systems(ed. R.A. Meyers), Springer, (2011), 12121236. 
[5] 
T. Ouyang, and Z. Xie, A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4body problem, preprint,, , (). 
[6] 
T. Ouyang, and Z. Xie, A continuum of periodic solutions to the fourbody problem with various choices of masses, preprint,, , (). 
[7] 
D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical nbody problem, Invent. Math., 155 (2004), 305362. 
[8] 
M. Šuvakov and V. Dmitrašinović, Three classes of Newtonian threebody planar periodic orbits, Phy. Rev. Lett., 110 (2013), 114301. 
[9] 
R. Vanderbei, New orbits for the nbody problem, Ann. N.Y. Acad. Sci., 1017 (2004), 422433. 
[10] 
L. Bakker, T. Ouyang, D. Yan, S. Simmons and G. Roberts, Linear stability for some symmetric periodic simultaneous binary collision orbits in the fourbody problem, Celestial Mech. Dynam. Astronom., 108 (2010), 147164. 
[11] 
D. Yan, Existence and linear stability of the rhomboidal periodic orbit in the planar equal mass fourbody problem, J. Math. Anal. Appl. 388 (2012), 942951. 
[12] 
T. Ouyang, S. Simmons and D.Yan, Periodic solutions with singularities in two dimensions in the nbody problem, Rocky Mountain J. Math., 42 (2012), 16011614. 
[13] 
D. Yan, and T. Ouyang, New phenomena in the spatial isosceles threebody 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 equalmass threebody problem,, preprint., (). 
show all references
References:
[1] 
R. Broucke, Classification of periodic orbits in the four and fivebody problems, Ann. N.Y. Acad. Sci., 1017 (2004), 408421. 
[2] 
K. Chen, Actionminimizing orbits in the parallelogram fourbody problem with equal masses, Arch. Ration. Mech. Anal., 170 (2001), 293318. 
[3] 
K. Chen, Variational methods on periodic and quasiperiodic solutions for the Nbody problem, Erg. Thy. Dyn. Sys., 23 (2003), 16911715. 
[4] 
L. Sbano, Periodic orbits of Hamiltonian systems, in Mathematics of Complexity and Dynamical Systems(ed. R.A. Meyers), Springer, (2011), 12121236. 
[5] 
T. Ouyang, and Z. Xie, A new variational method with SPBC and many stable choreographic solutions of the Newtonian 4body problem, preprint,, , (). 
[6] 
T. Ouyang, and Z. Xie, A continuum of periodic solutions to the fourbody problem with various choices of masses, preprint,, , (). 
[7] 
D. Ferrario and S. Terracini, On the existence of collisionless equivariant minimizers for the classical nbody problem, Invent. Math., 155 (2004), 305362. 
[8] 
M. Šuvakov and V. Dmitrašinović, Three classes of Newtonian threebody planar periodic orbits, Phy. Rev. Lett., 110 (2013), 114301. 
[9] 
R. Vanderbei, New orbits for the nbody problem, Ann. N.Y. Acad. Sci., 1017 (2004), 422433. 
[10] 
L. Bakker, T. Ouyang, D. Yan, S. Simmons and G. Roberts, Linear stability for some symmetric periodic simultaneous binary collision orbits in the fourbody problem, Celestial Mech. Dynam. Astronom., 108 (2010), 147164. 
[11] 
D. Yan, Existence and linear stability of the rhomboidal periodic orbit in the planar equal mass fourbody problem, J. Math. Anal. Appl. 388 (2012), 942951. 
[12] 
T. Ouyang, S. Simmons and D.Yan, Periodic solutions with singularities in two dimensions in the nbody problem, Rocky Mountain J. Math., 42 (2012), 16011614. 
[13] 
D. Yan, and T. Ouyang, New phenomena in the spatial isosceles threebody 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 equalmass threebody problem,, preprint., (). 
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