$P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R2n

Duanzhi Zhang

doi:10.3934/dcds.2013.33.947

Previous Article
DCDS Home
This Issue
Next Article
Traveling fronts and entire solutions in partially degenerate reaction-diffusion systems with monostable nonlinearity

February 2013, 33(2): 947-964. doi: 10.3934/dcds.2013.33.947

$P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R²ⁿ

Duanzhi Zhang ^1,

1.	School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China

Received July 2011 Revised November 2011 Published September 2012

Abstract
Related Papers

Let Σ be a $C^2$ compact strictly convex hypersurface in R²ⁿ with $n\ge 2$. Suppose $PΣ=Σ$ with $P$ being a $2n\times 2n$ symplectic and orthogonal matrix and $P^r=I_{2n}$. We prove that there are at least two geometrically distinct $P$-cyclic symmetric closed characteristics $(\tau_j,x_j)$ on Σ in the sense that $x_j(t+\frac{\tau_j}{r})=Px_j(t)$ for all $t∈R$ with $j=1,2$. As a corollary we obtain the existence of two geometrically distinct central symmetric closed characteristics on any $C^2$ central symmetric compact convex hypersurface in R²ⁿ with $n\ge 2$.

Keywords: dual variation., Maslov-type index, convex Hamiltonian systems, closed characteristics, $P$-cyclic symmetric.

Mathematics Subject Classification: Primary: 58E05, 70H05; Secondary: 34C2.

Citation: Duanzhi Zhang. $P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R²ⁿ. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 947-964. doi: 10.3934/dcds.2013.33.947

References:

[1]	S. E. Cappell, R. Lee and E. Y. Miller, On the Maslov-type index,, Comm. Pure Appl. Math, 47 (1994), 121. Google Scholar
[2]	C. Conley and E. Zehnder, Maslov-type index theory for flows and periodix solutions for Hamiltonian equations,, Commu. Pure. Appl. Math, 37 (1984), 207. doi: 10.1002/cpa.3160370204. Google Scholar
[3]	Y. Dong and Y. Long, Closed characteristics on partially symmetric compact convex hyper surfaces in R²ⁿ,, J. Differential Equations, 196 (2004), 226. doi: 10.1016/S0022-0396(03)00168-2. Google Scholar
[4]	Y. Dong, P-index theory for linear Hamiltonian systems and multiple solutions for nonlinear Hamiltonian systems,, Nonlinearity, 19 (2006), 1275. doi: 10.1088/0951-7715/19/6/004. Google Scholar
[5]	J. J. Duistermaat, "Fourier Integral Operators,", Birkhäauser, (1996). Google Scholar
[6]	I. Ekeland, "Convexity Methods in Hamiltonian Mechanics,", Spring-Verlag, (1990). Google Scholar
[7]	I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and there closed trajectories,, Comm. Math. Phys., 113 (1987), 419. doi: 10.1007/BF01221255. Google Scholar
[8]	I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées de systèmes hamiltoniens convexes,, Ann. Inst. H. Poincaré Anal. NonLinéaire, 4 (1987), 307. Google Scholar
[9]	E. Fadell and P. H. Rabinowitz, Generalized cohomologyical index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems,, Ivent. Math, 45 (1978), 139. Google Scholar
[10]	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. doi: 10.1007/s00220-009-0860-y. Google Scholar
[11]	C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hyper surfaces in R²ⁿ,, Math. Ann., 323 (2002), 201. Google Scholar
[12]	C. Liu and D. Zhang, Iteration theory of L-index and Multiplicity of brake orbits,, , (). Google Scholar
[13]	H. Liu, P-invariant closed characteristics on partially symmetric compact convex hypersurfsces in R²ⁿ., preprint, (). Google Scholar
[14]	Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains,, Advances in Math, 203 (2006), 568. Google Scholar
[15]	Y. Long, Bott formula of the Maslov-type index theory,, Pacific J. Math, 187 (1999), 113. Google Scholar
[16]	Y. Long, "Index Theory for Symplectic Paths with Applications,", Birkhäuser, (2002). Google Scholar
[17]	Y. Long and C. Zhu, Maslov-type index theory for symplectic paths and spectral flow(II),, Chinese Ann. of Math, 21 (2000), 89. doi: 10.1142/S0252959900000133. Google Scholar
[18]	Y. Long and C. Zhu, Closed characteristics on compact convex hyper surface in R²ⁿ,, Ann. of Math, 155 (2002), 317. Google Scholar
[19]	P. H. Rabinowitz, Periodic solution of Hamiltonian systems,, Commu. Pure Appl. Math, 31 (1978), 157. doi: 10.1002/cpa.3160310203. Google Scholar
[20]	J. Robbin and D. Salamon, The Maslov indices for paths,, Topology, 32 (1993), 827. doi: 10.1016/0040-9383(93)90052-W. Google Scholar
[21]	J. Robbin and D. Salamon, The spectral flow and the Maslov index,, Bull. London Math. Soc., 27 (1995), 1. doi: 10.1112/blms/27.1.1. Google Scholar
[22]	A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems,, Bull. Soc. Math. France, 116 (1988), 171. Google Scholar
[23]	K. Uhlenbeck, The Morse index theorem in Hilbert space,, J. Differential Geom., 8 (1973), 555. Google Scholar
[24]	W. Wang, X. Hu and Y. Long, Resonance identity, stability, and multiplicity of closed characteristics on compact convex hyper surfaces,, Duke Math. J., 139 (2007), 411. doi: 10.1215/S0012-7094-07-13931-0. Google Scholar
[25]	A. Weinstein, Periodic orbits for convex Hamiltonian systems,, Ann. of Math, 108 (1978), 507. doi: 10.2307/1971185. Google Scholar
[26]	D. Zhang, Multiple symmetric brake orbits in bounded convex symmetric domains,, Advanced Nonl. Studies, 6 (2006), 643. Google Scholar
[27]	C. Zhu and Y. Long, Maslov index theory for symplectic paths and spectral flow(I),, Chinese Ann. of Math, 208 (1999), 413. doi: 10.1142/S0252959999000485. Google Scholar

show all references

References:

[1]	S. E. Cappell, R. Lee and E. Y. Miller, On the Maslov-type index,, Comm. Pure Appl. Math, 47 (1994), 121. Google Scholar
[2]	C. Conley and E. Zehnder, Maslov-type index theory for flows and periodix solutions for Hamiltonian equations,, Commu. Pure. Appl. Math, 37 (1984), 207. doi: 10.1002/cpa.3160370204. Google Scholar
[3]	Y. Dong and Y. Long, Closed characteristics on partially symmetric compact convex hyper surfaces in R²ⁿ,, J. Differential Equations, 196 (2004), 226. doi: 10.1016/S0022-0396(03)00168-2. Google Scholar
[4]	Y. Dong, P-index theory for linear Hamiltonian systems and multiple solutions for nonlinear Hamiltonian systems,, Nonlinearity, 19 (2006), 1275. doi: 10.1088/0951-7715/19/6/004. Google Scholar
[5]	J. J. Duistermaat, "Fourier Integral Operators,", Birkhäauser, (1996). Google Scholar
[6]	I. Ekeland, "Convexity Methods in Hamiltonian Mechanics,", Spring-Verlag, (1990). Google Scholar
[7]	I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and there closed trajectories,, Comm. Math. Phys., 113 (1987), 419. doi: 10.1007/BF01221255. Google Scholar
[8]	I. Ekeland and L. Lassoued, Multiplicité des trajectoires fermées de systèmes hamiltoniens convexes,, Ann. Inst. H. Poincaré Anal. NonLinéaire, 4 (1987), 307. Google Scholar
[9]	E. Fadell and P. H. Rabinowitz, Generalized cohomologyical index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems,, Ivent. Math, 45 (1978), 139. Google Scholar
[10]	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. doi: 10.1007/s00220-009-0860-y. Google Scholar
[11]	C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hyper surfaces in R²ⁿ,, Math. Ann., 323 (2002), 201. Google Scholar
[12]	C. Liu and D. Zhang, Iteration theory of L-index and Multiplicity of brake orbits,, , (). Google Scholar
[13]	H. Liu, P-invariant closed characteristics on partially symmetric compact convex hypersurfsces in R²ⁿ., preprint, (). Google Scholar
[14]	Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains,, Advances in Math, 203 (2006), 568. Google Scholar
[15]	Y. Long, Bott formula of the Maslov-type index theory,, Pacific J. Math, 187 (1999), 113. Google Scholar
[16]	Y. Long, "Index Theory for Symplectic Paths with Applications,", Birkhäuser, (2002). Google Scholar
[17]	Y. Long and C. Zhu, Maslov-type index theory for symplectic paths and spectral flow(II),, Chinese Ann. of Math, 21 (2000), 89. doi: 10.1142/S0252959900000133. Google Scholar
[18]	Y. Long and C. Zhu, Closed characteristics on compact convex hyper surface in R²ⁿ,, Ann. of Math, 155 (2002), 317. Google Scholar
[19]	P. H. Rabinowitz, Periodic solution of Hamiltonian systems,, Commu. Pure Appl. Math, 31 (1978), 157. doi: 10.1002/cpa.3160310203. Google Scholar
[20]	J. Robbin and D. Salamon, The Maslov indices for paths,, Topology, 32 (1993), 827. doi: 10.1016/0040-9383(93)90052-W. Google Scholar
[21]	J. Robbin and D. Salamon, The spectral flow and the Maslov index,, Bull. London Math. Soc., 27 (1995), 1. doi: 10.1112/blms/27.1.1. Google Scholar
[22]	A. Szulkin, Morse theory and existence of periodic solutions of convex Hamiltonian systems,, Bull. Soc. Math. France, 116 (1988), 171. Google Scholar
[23]	K. Uhlenbeck, The Morse index theorem in Hilbert space,, J. Differential Geom., 8 (1973), 555. Google Scholar
[24]	W. Wang, X. Hu and Y. Long, Resonance identity, stability, and multiplicity of closed characteristics on compact convex hyper surfaces,, Duke Math. J., 139 (2007), 411. doi: 10.1215/S0012-7094-07-13931-0. Google Scholar
[25]	A. Weinstein, Periodic orbits for convex Hamiltonian systems,, Ann. of Math, 108 (1978), 507. doi: 10.2307/1971185. Google Scholar
[26]	D. Zhang, Multiple symmetric brake orbits in bounded convex symmetric domains,, Advanced Nonl. Studies, 6 (2006), 643. Google Scholar
[27]	C. Zhu and Y. Long, Maslov index theory for symplectic paths and spectral flow(I),, Chinese Ann. of Math, 208 (1999), 413. doi: 10.1142/S0252959999000485. Google Scholar

[1]	Hui Liu, Duanzhi Zhang. Stable P-symmetric closed characteristics on partially symmetric compact convex hypersurfaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 877-893. doi: 10.3934/dcds.2016.36.877
[2]	Wei Wang. Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 679-701. doi: 10.3934/dcds.2012.32.679
[3]	Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 887-912. doi: 10.3934/dcdsb.2018047
[4]	Somphong Jitman, San Ling, Ekkasit Sangwisut. On self-dual cyclic codes of length $p^a$ over $GR(p^2,s)$. Advances in Mathematics of Communications, 2016, 10 (2) : 255-273. doi: 10.3934/amc.2016004
[5]	Morched Boughariou. Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 603-616. doi: 10.3934/dcds.2003.9.603
[6]	Cem Güneri, Ferruh Özbudak, Funda ÖzdemIr. On complementary dual additive cyclic codes. Advances in Mathematics of Communications, 2017, 11 (2) : 353-357. doi: 10.3934/amc.2017028
[7]	Renato Manfrin. On the global solvability of symmetric hyperbolic systems of Kirchhoff type. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 91-106. doi: 10.3934/dcds.1997.3.91
[8]	Jianshe Yu, Honghua Bin, Zhiming Guo. Periodic solutions for discrete convex Hamiltonian systems via Clarke duality. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 939-950. doi: 10.3934/dcds.2006.15.939
[9]	Giuseppe Cordaro. Existence and location of periodic solutions to convex and non coercive Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 983-996. doi: 10.3934/dcds.2005.12.983
[10]	André Vanderbauwhede. Continuation and bifurcation of multi-symmetric solutions in reversible Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 359-363. doi: 10.3934/dcds.2013.33.359
[11]	Chungen Liu. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 337-355. doi: 10.3934/dcds.2010.27.337
[12]	Rumei Zhang, Jin Chen, Fukun Zhao. Multiple solutions for superlinear elliptic systems of Hamiltonian type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1249-1262. doi: 10.3934/dcds.2011.30.1249
[13]	Hugo Beirão da Veiga, Francesca Crispo. On the global regularity for nonlinear systems of the $p$-Laplacian type. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1173-1191. doi: 10.3934/dcdss.2013.6.1173
[14]	Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$-Laplacian systems of Liénard-type. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1393-1400. doi: 10.3934/cpaa.2011.10.1393
[15]	Shenzhou Zheng, Laping Zhang, Zhaosheng Feng. Everywhere regularity for P-harmonic type systems under the subcritical growth. Communications on Pure & Applied Analysis, 2008, 7 (1) : 107-117. doi: 10.3934/cpaa.2008.7.107
[16]	Long Yu, Hongwei Liu. A class of $p$-ary cyclic codes and their weight enumerators. Advances in Mathematics of Communications, 2016, 10 (2) : 437-457. doi: 10.3934/amc.2016017
[17]	Salvatore A. Marano, Sunra Mosconi. Non-smooth critical point theory on closed convex sets. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1187-1202. doi: 10.3934/cpaa.2014.13.1187
[18]	Eric Bedford, Kyounghee Kim. Degree growth of matrix inversion: Birational maps of symmetric, cyclic matrices. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 977-1013. doi: 10.3934/dcds.2008.21.977
[19]	Norimichi Hirano, Wieslaw Krawcewicz, Haibo Ruan. Existence of nonstationary periodic solutions for $\Gamma$-symmetric Lotka-Volterra type systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 709-735. doi: 10.3934/dcds.2011.30.709
[20]	Xiao-Fei Zhang, Fei Guo. Multiplicity of subharmonic solutions and periodic solutions of a particular type of super-quadratic Hamiltonian systems. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1625-1642. doi: 10.3934/cpaa.2016005

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences

$P$-cyclic symmetric closed characteristics on compact convex $P$-cyclic symmetric hypersurface in R²ⁿ

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences