Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces

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February 2016, 36(2): 877-893. doi: 10.3934/dcds.2016.36.877

Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces

Hui Liu ^1, and Duanzhi Zhang ^2,

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

Received February 2014 Published August 2015

Abstract
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In this paper, let $n\geq2$ be an integer, $P=diag(-I_{n-\kappa},I_\kappa,-I_{n-\kappa}, I_\kappa)$ for some integer $\kappa\in[0, n-1)$, and $\Sigma \subset {\bf R}^{2n}$ be a partially symmetric compact convex hypersurface, i.e., $x\in \Sigma$ implies $Px\in\Sigma$. We prove that if $\Sigma$ is $(r,R)$-pinched with $\frac{R}{r}<\sqrt{\frac{5}{3}}$, then $\Sigma$ carries at least two geometrically distinct P-symmetric closed characteristics which possess at least $2n-4\kappa$ Floquet multipliers on the unit circle of the complex plane.

Keywords: Hamiltonian system., stable P-symmetric closed characteristics, Compact convex hypersurfaces, P-index iteration.

Mathematics Subject Classification: Primary: 58E05, 37J45; Secondary: 34C2.

Citation: Hui Liu, Duanzhi Zhang. Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 877-893. doi: 10.3934/dcds.2016.36.877

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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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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