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Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces
1. | Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, China |
2. | School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071 |
References:
[1] |
Y. Dong and Y. Long, Closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, J. Differential Equations, 196 (2004), 226-248.
doi: 10.1016/S0022-0396(03)00168-2. |
[2] |
Y. Dong and Y. Long, Stable closed characteristics on partially symmetric compact convex hypersurfaces, J. Differential Equations, 206 (2004), 265-279.
doi: 10.1016/j.jde.2004.03.004. |
[3] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag. Berlin. 1990.
doi: 10.1007/978-3-642-74331-3. |
[4] |
I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories, Comm. Math. Phys., 113 (1987), 419-469.
doi: 10.1007/BF01221255. |
[5] |
I. Ekeland and J. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math., 112 (1980), 283-319.
doi: 10.2307/1971148. |
[6] |
I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées d'un systéme hamiltonien sur une hypersurface d'energie convexe, Ann. IHP. Anal. non Linéaire., 4 (1987), 307-335. |
[7] |
E. Fadell and P. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation equations for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.
doi: 10.1007/BF01390270. |
[8] |
M. Girardi, Multiple orbits for Hamiltonian systems on starshaped ernergy surfaces with symmetry, Ann. IHP. Analyse non linéaire., 1 (1984), 285-294. |
[9] |
X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltoinan systems with application to figure-eight orbits, Commun. Math. Phys., 290 (2009), 737-777.
doi: 10.1007/s00220-009-0860-y. |
[10] |
H. Liu, Stability of symmetric closed characteristics on symmetric compact convex hypersurfaces in $R^{2n}$ under a pinching condition, Acta Mathematica Sinica, English Series, 28 (2012), 885-900.
doi: 10.1007/s10114-011-0494-9. |
[11] |
H. Liu, Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, Cal. Variations and PDEs, 49 (2014), 1121-1147.
doi: 10.1007/s00526-013-0614-8. |
[12] |
H. Liu and D. Zhang, On the number of P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, Science China Mathematics, 58 (2015), 1771-1778. |
[13] |
C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $R^{2n}$, Math. Ann., 323 (2002), 201-215.
doi: 10.1007/s002089100257. |
[14] |
Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkhäuser. Basel. 2002.
doi: 10.1007/978-3-0348-8175-3. |
[15] |
Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $R^{2n}$, Ann. of Math., 155 (2002), 317-368.
doi: 10.2307/3062120. |
[16] |
P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
[17] |
A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France., 116 (1988), 171-197. |
[18] |
W. Wang, Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete Contin. Dyn. Syst., 32 (2012), 679-701.
doi: 10.3934/dcds.2012.32.679. |
[19] |
A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.
doi: 10.2307/1971185. |
[20] |
W. Wang, X. Hu and Y. Long, Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462.
doi: 10.1215/S0012-7094-07-13931-0. |
[21] |
D. Zhang, P-cyclic symmetric closed characteristics on compact convex P-cyclic symmetric hypersurface in $R^{2n}$, Discrete Contin. Dyn. Syst., 33 (2013), 947-964.
doi: 10.3934/dcds.2013.33.947. |
show all references
References:
[1] |
Y. Dong and Y. Long, Closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, J. Differential Equations, 196 (2004), 226-248.
doi: 10.1016/S0022-0396(03)00168-2. |
[2] |
Y. Dong and Y. Long, Stable closed characteristics on partially symmetric compact convex hypersurfaces, J. Differential Equations, 206 (2004), 265-279.
doi: 10.1016/j.jde.2004.03.004. |
[3] |
I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag. Berlin. 1990.
doi: 10.1007/978-3-642-74331-3. |
[4] |
I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their closed trajectories, Comm. Math. Phys., 113 (1987), 419-469.
doi: 10.1007/BF01221255. |
[5] |
I. Ekeland and J. Lasry, On the number of periodic trajectories for a Hamiltonian flow on a convex energy surface, Ann. of Math., 112 (1980), 283-319.
doi: 10.2307/1971148. |
[6] |
I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées d'un systéme hamiltonien sur une hypersurface d'energie convexe, Ann. IHP. Anal. non Linéaire., 4 (1987), 307-335. |
[7] |
E. Fadell and P. Rabinowitz, Generalized cohomological index theories for Lie group actions with an application to bifurcation equations for Hamiltonian systems, Invent. Math., 45 (1978), 139-174.
doi: 10.1007/BF01390270. |
[8] |
M. Girardi, Multiple orbits for Hamiltonian systems on starshaped ernergy surfaces with symmetry, Ann. IHP. Analyse non linéaire., 1 (1984), 285-294. |
[9] |
X. Hu and S. Sun, Index and stability of symmetric periodic orbits in Hamiltoinan systems with application to figure-eight orbits, Commun. Math. Phys., 290 (2009), 737-777.
doi: 10.1007/s00220-009-0860-y. |
[10] |
H. Liu, Stability of symmetric closed characteristics on symmetric compact convex hypersurfaces in $R^{2n}$ under a pinching condition, Acta Mathematica Sinica, English Series, 28 (2012), 885-900.
doi: 10.1007/s10114-011-0494-9. |
[11] |
H. Liu, Multiple P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, Cal. Variations and PDEs, 49 (2014), 1121-1147.
doi: 10.1007/s00526-013-0614-8. |
[12] |
H. Liu and D. Zhang, On the number of P-invariant closed characteristics on partially symmetric compact convex hypersurfaces in $R^{2n}$, Science China Mathematics, 58 (2015), 1771-1778. |
[13] |
C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in $R^{2n}$, Math. Ann., 323 (2002), 201-215.
doi: 10.1007/s002089100257. |
[14] |
Y. Long, Index Theory for Symplectic Paths with Applications, Progress in Math. 207, Birkhäuser. Basel. 2002.
doi: 10.1007/978-3-0348-8175-3. |
[15] |
Y. Long and C. Zhu, Closed characteristics on compact convex hypersurfaces in $R^{2n}$, Ann. of Math., 155 (2002), 317-368.
doi: 10.2307/3062120. |
[16] |
P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184.
doi: 10.1002/cpa.3160310203. |
[17] |
A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France., 116 (1988), 171-197. |
[18] |
W. Wang, Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete Contin. Dyn. Syst., 32 (2012), 679-701.
doi: 10.3934/dcds.2012.32.679. |
[19] |
A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518.
doi: 10.2307/1971185. |
[20] |
W. Wang, X. Hu and Y. Long, Resonance identity, stability and multiplicity of closed characteristics on compact convex hypersurfaces, Duke Math. J., 139 (2007), 411-462.
doi: 10.1215/S0012-7094-07-13931-0. |
[21] |
D. Zhang, P-cyclic symmetric closed characteristics on compact convex P-cyclic symmetric hypersurface in $R^{2n}$, Discrete Contin. Dyn. Syst., 33 (2013), 947-964.
doi: 10.3934/dcds.2013.33.947. |
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