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Stability of the rhomboidal symmetric-mass orbit
1. | 275 TMCB, Brigham Young University, Provo, UT 84602, United States, United States |
References:
[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.
doi: 10.1016/j.jmaa.2012.03.022. |
[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.
doi: 10.1007/s10569-011-9358-y. |
[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.
doi: 10.1007/s10569-012-9402-6. |
[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.
doi: 10.1007/s10569-010-9298-y. |
[5] |
A. Bounemoura, Generic super-exponential stability of invariant tori in Hamiltonian systems, Ergodic Theory Dynam. Systems, 31 (2011), 1287-1303.
doi: 10.1017/S0143385710000441. |
[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.
doi: 10.1063/1.165984. |
[8] |
Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, 207, Birkhäuser Verlag, Basel, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[9] |
R. Martínez, On the existence of doubly symmetric "Schubart-like'' periodic orbits, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 943-975.
doi: 10.3934/dcdsb.2012.17.943. |
[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.
doi: 10.3934/dcdsb.2008.10.609. |
[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.
doi: 10.1007/s10569-010-9325-z. |
[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.
doi: 10.1216/RMJ-2012-42-5-1601. |
[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.
doi: 10.1017/S0143385707000284. |
[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.
doi: 10.1023/A:1011102815021. |
[16] |
J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22.
doi: 10.1002/asna.19562830105. |
[17] |
M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n$-body problem, Arch. Ration. Mech. Anal., 199 (2011), 821-841.
doi: 10.1007/s00205-010-0334-6. |
[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.
doi: 10.1007/s10569-010-9270-x. |
[21] |
W. L. Sweatman, The symmetrical one-dimensional Newtonian four-body problem: A numerical investigation, Celestial Mech. Dynam. Astronom., 82 (2002), 179-201.
doi: 10.1023/A:1014599918133. |
[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.
doi: 10.1007/s10569-005-2289-8. |
[23] |
A. Venturelli, A variational proof of the existence of von Schubart's orbit, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 699-717.
doi: 10.3934/dcdsb.2008.10.699. |
[24] |
J. Waldvogel, The rhomboidal symmetric four-body problem, Celestial Mech. Dynam. Astronom., 113 (2012), 113-123.
doi: 10.1007/s10569-012-9414-2. |
[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.
doi: 10.1016/j.jmaa.2011.10.032. |
show all references
References:
[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.
doi: 10.1016/j.jmaa.2012.03.022. |
[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.
doi: 10.1007/s10569-011-9358-y. |
[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.
doi: 10.1007/s10569-012-9402-6. |
[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.
doi: 10.1007/s10569-010-9298-y. |
[5] |
A. Bounemoura, Generic super-exponential stability of invariant tori in Hamiltonian systems, Ergodic Theory Dynam. Systems, 31 (2011), 1287-1303.
doi: 10.1017/S0143385710000441. |
[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.
doi: 10.1063/1.165984. |
[8] |
Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Mathematics, 207, Birkhäuser Verlag, Basel, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[9] |
R. Martínez, On the existence of doubly symmetric "Schubart-like'' periodic orbits, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 943-975.
doi: 10.3934/dcdsb.2012.17.943. |
[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.
doi: 10.3934/dcdsb.2008.10.609. |
[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.
doi: 10.1007/s10569-010-9325-z. |
[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.
doi: 10.1216/RMJ-2012-42-5-1601. |
[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.
doi: 10.1017/S0143385707000284. |
[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.
doi: 10.1023/A:1011102815021. |
[16] |
J. Schubart, Numerische Aufsuchung periodischer Lösungen im Dreikörperproblem, Astr. Nachr., 283 (1956), 17-22.
doi: 10.1002/asna.19562830105. |
[17] |
M. Shibayama, Minimizing periodic orbits with regularizable collisions in the $n$-body problem, Arch. Ration. Mech. Anal., 199 (2011), 821-841.
doi: 10.1007/s00205-010-0334-6. |
[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.
doi: 10.1007/s10569-010-9270-x. |
[21] |
W. L. Sweatman, The symmetrical one-dimensional Newtonian four-body problem: A numerical investigation, Celestial Mech. Dynam. Astronom., 82 (2002), 179-201.
doi: 10.1023/A:1014599918133. |
[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.
doi: 10.1007/s10569-005-2289-8. |
[23] |
A. Venturelli, A variational proof of the existence of von Schubart's orbit, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 699-717.
doi: 10.3934/dcdsb.2008.10.699. |
[24] |
J. Waldvogel, The rhomboidal symmetric four-body problem, Celestial Mech. Dynam. Astronom., 113 (2012), 113-123.
doi: 10.1007/s10569-012-9414-2. |
[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.
doi: 10.1016/j.jmaa.2011.10.032. |
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