American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[3]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag. Berlin. 1990. doi: 10.1007/978-3-642-74331-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Girardi, Multiple orbits for Hamiltonian systems on starshaped ernergy surfaces with symmetry, Ann. IHP. Analyse non linéaire., 1 (1984), 285-294.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.  Google Scholar

[17]

A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France., 116 (1988), 171-197.  Google Scholar

[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.  Google Scholar

[19]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518. doi: 10.2307/1971185.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

I. Ekeland, Convexity Methods in Hamiltonian Mechanics, Springer-Verlag. Berlin. 1990. doi: 10.1007/978-3-642-74331-3.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

M. Girardi, Multiple orbits for Hamiltonian systems on starshaped ernergy surfaces with symmetry, Ann. IHP. Analyse non linéaire., 1 (1984), 285-294.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

P. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure. Appl. Math., 31 (1978), 157-184. doi: 10.1002/cpa.3160310203.  Google Scholar

[17]

A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems, Bull. Soc. Math. France., 116 (1988), 171-197.  Google Scholar

[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.  Google Scholar

[19]

A. Weinstein, Periodic orbits for convex Hamiltonian systems, Ann. of Math., 108 (1978), 507-518. doi: 10.2307/1971185.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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