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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

Stability of the rhomboidal symmetric-mass orbit
Pages: 1 - 23, Issue 1, January 2015

doi:10.3934/dcds.2015.35.1      Abstract        References        Full text (958.8K)           Related Articles

Lennard Bakker - 275 TMCB, Brigham Young University, Provo, UT 84602, United States (email)
Skyler Simmons - 275 TMCB, Brigham Young University, Provo, UT 84602, United States (email)

1 L. F. Bakker, S. Mancuso and S. C. Simmons, Linear stability for some symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem, J. Math. Anal. Appl., 392 (2012), 136-147.       
2 L. F. Bakker, T. Ouyang, D. Yan and S. Simmons, Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem, Celestial Mech. Dynam. Astronom., 110 (2011), 271-290.       
3 L. F. Bakker, T. Ouyang, D. Yan and S. Simmons, Erratum to: Existence and stability of symmetric periodic simultaneous binary collision orbits in the planar pairwise symmetric four-body problem [mr2821623], Celestial Mech. Dynam. Astronom., 112 (2012), 459-460.       
4 L. F. Bakker, T. Ouyang, D. Yan, S. Simmons and G. E. Roberts, Linear stability for some symmetric periodic simultaneous binary collision orbits in the four-body problem, Celestial Mech. Dynam. Astronom., 108 (2010), 147-164.       
5 A. Bounemoura, Generic super-exponential stability of invariant tori in Hamiltonian systems, Ergodic Theory Dynam. Systems, 31 (2011), 1287-1303.       
6 M. Hénon, Stability of interplay oribts, Cel. Mech., 15 (1977), 243-261.
7 J. Hietarinta and S. Mikkola, Chaos in the one-dimensional gravitational three-body problem, Chaos, 3 (1993), 183-203.       
8 Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, 207, Birkhäuser Verlag, Basel, 2002.       
9 R. Martínez, On the existence of doubly symmetric "Schubart-like'' periodic orbits, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 943-975.       
10 K. R. Meyer, G. R. Hall and D. Offin, Introduction to Hamiltonian Dynamical Systems and the $N$-body Problem, 2nd edition, Applied Mathematical Sciences, 90, Springer, New York, 2009.       
11 R. Moeckel, A topological existence proof for the Schubart orbits in the collinear three-body problem, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 609-620.       
12 T. Ouyang and D. Yan, Periodic solutions with alternating singularities in the collinear four-body problem, Celestial Mech. Dynam. Astronom., 109 (2011), 229-239.       
13 T. Ouyang, D. Yan and S. Simmons, Periodic solutions with singularities in two dimensions in the $n$-body problem, Rocky Mtn. J. Math., 42 (2012), 1601-1614.       
14 G. E. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergodic Theory Dynam. Systems, 27 (2007), 1947-1963.       
15 A. E. Roy and B. A. Steves, The Caledonian symmetrical double binary four-body problem. I. Surfaces of zero-velocity using the energy integral, Celestial Mech. Dynam. Astronom., 78 (2000), 299-318.       
16 J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22.       
17 M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n$-body problem, Arch. Ration. Mech. Anal., 199 (2011), 821-841.       
18 C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 1995.       
19 C. Simó, New families of solutions in $N$-body problems, in European Congress of Mathematics, Vol. I (Barcelona, 2000), Progr. Math., 201, Birkhäuser, Basel, 2001, 101-115.       
20 A. Sivasankaran, B. A. Steves and W. L. Sweatman, A global regularisation for integrating the Caledonian symmetric four-body problem, Celestial Mech. Dynam. Astronom., 107 (2010), 157-168.       
21 W. L. Sweatman, The symmetrical one-dimensional Newtonian four-body problem: A numerical investigation, Celestial Mech. Dynam. Astronom., 82 (2002), 179-201.       
22 W. L. Sweatman, A family of symmetrical Schubart-like interplay orbits and their stability in the one-dimensional four-body problem, Celestial Mech. Dynam. Astronom., 94 (2006), 37-65.       
23 A. Venturelli, A variational proof of the existence of von Schubart's orbit, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 699-717.       
24 J. Waldvogel, The rhomboidal symmetric four-body problem, Celestial Mech. Dynam. Astronom., 113 (2012), 113-123.       
25 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.       

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