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A class of adding machines and Julia sets
Linear stability of the criss-cross orbit in the equal-mass three-body problem
1. | School of Mathematics and Statistics, & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024 |
2. | Department of Mathematics, Brigham Young University, Provo, Utah 84602 |
3. | School of Mathematics and System Science, Beihang University, Beijing 100191 |
References:
[1] |
V. Barutello, R. Jadanza and A. Portaluri, Linear instability of relative equilibria for n-body problems in the plane, J. Diff. Eqn., 257 (2014), 1773-1813.
doi: 10.1016/j.jde.2014.05.017. |
[2] |
V. Barutello, R. Jadanza and A. Portaluri, Morse index and linear stability of the Lagrangian circular orbit in a three-body-type problem via index theory, Arch. Ration. Mech. Anal., 219 (2016), 387-444.
doi: 10.1007/s00205-015-0898-2. |
[3] |
R. Broucke, On relative periodic solutions of the planar general three-body problem, Celestial Mech., 12 (1975), 439-462.
doi: 10.1007/BF01595390. |
[4] |
S. Cappell, R. Lee and E. Miller, On the Maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186.
doi: 10.1002/cpa.3160470202. |
[5] |
A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math., 152 (2000), 881-901.
doi: 10.2307/2661357. |
[6] |
A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing 2002), 279-294, Higher Ed. Press, Beijing, 2002. |
[7] |
K.-C. Chen, T. Ouyang and Z. Xia, Action-minimizing periodic and quasi-periodic solutions in the n-body problem, Math. Res. Lett., 19 (2012), 483-497.
doi: 10.4310/MRL.2012.v19.n2.a19. |
[8] |
K.-C. Chen, Existence and minimizing properties of retrograde orbits to the three-body problems with various choices of masses, Ann. of Math., 167 (2008), 325-348.
doi: 10.4007/annals.2008.167.325. |
[9] |
K.-C. Chen and Y. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Commun. Math. Phys., 291 (2009), 403-441.
doi: 10.1007/s00220-009-0769-5. |
[10] |
D. Ferrario and S. Terracini, On the existence of collisionless equivalent minimizers for the classical n-body problem, Invent. Math., 155 (2004), 305-362.
doi: 10.1007/s00222-003-0322-7. |
[11] |
M. Hénon, A family of periodic orbits of the planar three-body problem, and their atability, Celest. Mech. Dynam. Astron., 13 (1976), 267-285.
doi: 10.1007/BF01228647. |
[12] |
X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltonian system with application to Figure-eight orbit, Commun. Math. Phys., 290 (2009), 737-777.
doi: 10.1007/s00220-009-0860-y. |
[13] |
X. Hu and S. Sun, Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem, Adv. Math., 223 (2010), 98-119.
doi: 10.1016/j.aim.2009.07.017. |
[14] |
X. Hu, Y. Long and S. Sun, Linear stability of elliptic Lagrangian solutions of the classical planar three-body problem via index theory, Arch. Ration. Mech. Anal., 213 (2014), 993-1045.
doi: 10.1007/s00205-014-0749-6. |
[15] |
T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Berlin-Heidelberg-New York, Spring-Verlag, 1995. |
[16] |
Y. Long, Index Theory For Symplectic Paths With Applications, Birkhäuser Verlag, Basel-Boston-Berlin, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[17] |
C. Marchal, How the method of minimization of action avoids singularities, Celestial Mech. Dynam. Astronom., 83 (2002), 325-353.
doi: 10.1023/A:1020128408706. |
[18] |
C. Moore, Braids in classical gravity, Phys. Rev. Lett., 70 (1993), 3675-3679.
doi: 10.1103/PhysRevLett.70.3675. |
[19] |
C. Moore, , http://www.santafe.edu/textasciitilde moore/gallery.html., ().
|
[20] |
C. Moore and M. Nauenberg, New periodic orbits for the n-body problem, J. of Comput. Nonlin. Dyn., 1 (2006), 307-311.
doi: 10.1115/1.2338323. |
[21] |
T. Ouyang and Z. Xie, Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76.
doi: 10.1016/j.physd.2015.05.015. |
[22] |
G. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergod. Th. & Dynam. Sys., 27 (2007), 1947-1963.
doi: 10.1017/S0143385707000284. |
[23] |
C. Zhu, A generalized Morse index theorem, In Analysis, Geometry and Topology of Elliptic Operators, Hackensack, NJ: World Sci. Publ. (2006), 493-540. |
[24] |
G. Zhu and Y. Long, Linear stability of some symplectic matrices, Front. Math. China, 5 (2010), 361-368.
doi: 10.1007/s11464-010-0008-6. |
show all references
References:
[1] |
V. Barutello, R. Jadanza and A. Portaluri, Linear instability of relative equilibria for n-body problems in the plane, J. Diff. Eqn., 257 (2014), 1773-1813.
doi: 10.1016/j.jde.2014.05.017. |
[2] |
V. Barutello, R. Jadanza and A. Portaluri, Morse index and linear stability of the Lagrangian circular orbit in a three-body-type problem via index theory, Arch. Ration. Mech. Anal., 219 (2016), 387-444.
doi: 10.1007/s00205-015-0898-2. |
[3] |
R. Broucke, On relative periodic solutions of the planar general three-body problem, Celestial Mech., 12 (1975), 439-462.
doi: 10.1007/BF01595390. |
[4] |
S. Cappell, R. Lee and E. Miller, On the Maslov index, Comm. Pure Appl. Math., 47 (1994), 121-186.
doi: 10.1002/cpa.3160470202. |
[5] |
A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Ann. of Math., 152 (2000), 881-901.
doi: 10.2307/2661357. |
[6] |
A. Chenciner, Action minimizing solutions of the Newtonian n-body problem: from homology to symmetry, Proceedings of the International Congress of Mathematicians, Vol. III (Beijing 2002), 279-294, Higher Ed. Press, Beijing, 2002. |
[7] |
K.-C. Chen, T. Ouyang and Z. Xia, Action-minimizing periodic and quasi-periodic solutions in the n-body problem, Math. Res. Lett., 19 (2012), 483-497.
doi: 10.4310/MRL.2012.v19.n2.a19. |
[8] |
K.-C. Chen, Existence and minimizing properties of retrograde orbits to the three-body problems with various choices of masses, Ann. of Math., 167 (2008), 325-348.
doi: 10.4007/annals.2008.167.325. |
[9] |
K.-C. Chen and Y. Lin, On action-minimizing retrograde and prograde orbits of the three-body problem, Commun. Math. Phys., 291 (2009), 403-441.
doi: 10.1007/s00220-009-0769-5. |
[10] |
D. Ferrario and S. Terracini, On the existence of collisionless equivalent minimizers for the classical n-body problem, Invent. Math., 155 (2004), 305-362.
doi: 10.1007/s00222-003-0322-7. |
[11] |
M. Hénon, A family of periodic orbits of the planar three-body problem, and their atability, Celest. Mech. Dynam. Astron., 13 (1976), 267-285.
doi: 10.1007/BF01228647. |
[12] |
X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltonian system with application to Figure-eight orbit, Commun. Math. Phys., 290 (2009), 737-777.
doi: 10.1007/s00220-009-0860-y. |
[13] |
X. Hu and S. Sun, Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem, Adv. Math., 223 (2010), 98-119.
doi: 10.1016/j.aim.2009.07.017. |
[14] |
X. Hu, Y. Long and S. Sun, Linear stability of elliptic Lagrangian solutions of the classical planar three-body problem via index theory, Arch. Ration. Mech. Anal., 213 (2014), 993-1045.
doi: 10.1007/s00205-014-0749-6. |
[15] |
T. Kato, Perturbation theory for linear operators, Classics in Mathematics, Berlin-Heidelberg-New York, Spring-Verlag, 1995. |
[16] |
Y. Long, Index Theory For Symplectic Paths With Applications, Birkhäuser Verlag, Basel-Boston-Berlin, 2002.
doi: 10.1007/978-3-0348-8175-3. |
[17] |
C. Marchal, How the method of minimization of action avoids singularities, Celestial Mech. Dynam. Astronom., 83 (2002), 325-353.
doi: 10.1023/A:1020128408706. |
[18] |
C. Moore, Braids in classical gravity, Phys. Rev. Lett., 70 (1993), 3675-3679.
doi: 10.1103/PhysRevLett.70.3675. |
[19] |
C. Moore, , http://www.santafe.edu/textasciitilde moore/gallery.html., ().
|
[20] |
C. Moore and M. Nauenberg, New periodic orbits for the n-body problem, J. of Comput. Nonlin. Dyn., 1 (2006), 307-311.
doi: 10.1115/1.2338323. |
[21] |
T. Ouyang and Z. Xie, Star pentagon and many stable choreographic solutions of the Newtonian 4-body problem, Physica D, 307 (2015), 61-76.
doi: 10.1016/j.physd.2015.05.015. |
[22] |
G. Roberts, Linear stability analysis of the figure-eight orbit in the three-body problem, Ergod. Th. & Dynam. Sys., 27 (2007), 1947-1963.
doi: 10.1017/S0143385707000284. |
[23] |
C. Zhu, A generalized Morse index theorem, In Analysis, Geometry and Topology of Elliptic Operators, Hackensack, NJ: World Sci. Publ. (2006), 493-540. |
[24] |
G. Zhu and Y. Long, Linear stability of some symplectic matrices, Front. Math. China, 5 (2010), 361-368.
doi: 10.1007/s11464-010-0008-6. |
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