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July  2019, 39(7): 4187-4206. doi: 10.3934/dcds.2019169

Brake orbits on compact symmetric dynamically convex reversible hypersurfaces on $ \mathbb{R}^\text{2n} $

Zhongjie Liu and  Duanzhi Zhang ,

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

* Corresponding author. Partially supported by the NSF of China (17190271, 11422103, 11771341) and Nankai University. E-mail: zhangdz@nankai.edu.cn

Received  October 2018 Revised  December 2018 Published  April 2019

In this paper, we consider multiplicity of brake orbits on compact symmetric dynamically convex reversible hypersurfaces in $\mathbb{R}^{2n}$. We prove that there exist at least \([\frac{n+1}{2}]\) geometrically distinct closed characteristics on dynamically convex hypersurface $\Sigma$ in $R^{2n}$ with the symmetric and reversible conditions, i.e. $\Sigma = -\Sigma$ and $N\Sigma = \Sigma$, where $N = diag(-I_{n}, I_{n})$. For $n\geq2$, we prove that there are at least 2 symmetric brake orbits on $\Sigma$, which generalizes Kang's result in [8] of 2013.

Keywords: Brake orbit, closed characteristic, dynamically convex hypersurface, Maslov-type index.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Zhongjie Liu, Duanzhi Zhang. Brake orbits on compact symmetric dynamically convex reversible hypersurfaces on $ \mathbb{R}^\text{2n} $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4187-4206. doi: 10.3934/dcds.2019169
References:
[1] A. Borel, Seminar on Transformation Groups, Princeton Univ. Press, Princeton, 1960.   Google Scholar
[2]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhauser, Basel, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[3]

H. Duan and H. Liu, Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in ${\Bbb R}^{2n}$, Calc. Var. Partial Differential Equations, 56 (2017), Art. 65, 30 pp, arXiv: 1601.03470. doi: 10.1007/s00526-017-1173-1.  Google Scholar

[4]

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

[5]

I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their periodic trajectories, Comm. Math. Phys, 113 (1987), 419-467.  doi: 10.1007/BF01221255.  Google Scholar

[6]

J. Gutt and J. Kang, On the minimal number of periodic orbits on some hypersurfaces in ${\Bbb R}^{2n}$, Ann. Inst. Fourier (Grenoble), 66 (2016), no. 6, 2485-2505. Google Scholar

[7]

C. Liu.-G., Maslov-type index theory for symplectic paths with Lagrangian boundary conditions, Adv. Nonlinear Stud, 7 (2007), 131-161.  doi: 10.1515/ans-2007-0107.  Google Scholar

[8]

J. Kang, Some remarks on symmetric periodic orbits in the restricted three-body problem, Discrete Contin. Dyn. Syst., 34 (2014), 5229-5245.  doi: 10.3934/dcds.2014.34.5229.  Google Scholar

[9]

C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in ${\Bbb R}^{2n}$, Math. Ann, 323 (2002), 201-215.  doi: 10.1007/s002089100257.  Google Scholar

[10]

C. Liu and D. Zhang, Seifert conjecture in the even convex case, Comm. Pure Appl. Math, 67 (2014), 1563-1604.  doi: 10.1002/cpa.21525.  Google Scholar

[11]

C. Liu and D. Zhang, Iteration theory of $L$-index and multiplicity of brake orbits, J. Differential Equations, 257 (2014), 1194–1245, arXiv: 0908.0021 [math.SG] doi: 10.1016/j.jde.2014.05.006.  Google Scholar

[12]

H. Liu, Y. Long and W. Wang, Resonance identities for closed characteristics on compact star-shaped hypersurfaces in ${\Bbb R}^2n$, J. Funct. Anal., 266 (2014), 5598-5638.  doi: 10.1016/j.jfa.2014.02.035.  Google Scholar

[13]

Y. Long, The topological structures of $\omega-subsets$ of symplectic groups, Acta Math., Sinica(English Series), 15 (1999), 255–268. doi: 10.1007/BF02650669.  Google Scholar

[14]

Y. Long, Index Theory for Symplectic Paths with Applictions, Progress in Mathematics. 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[15]

Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math, 187 (1999), 113-149.  doi: 10.2140/pjm.1999.187.113.  Google Scholar

[16]

Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Adv.of Math, 203 (2006), 568-635.  doi: 10.1016/j.aim.2005.05.005.  Google Scholar

[17]

C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math.Soc, 311 (1989), 621-655.  doi: 10.1090/S0002-9947-1989-0978370-5.  Google Scholar

[18]

C. Viterbo, Une théorie de Morse pour les systèmes hamiltoniens étoilés, C. R. Acad. Sci Paris. Ser. I Math, 301 (1985), 487–489.  Google Scholar

[19]

W. Wang, Closed trajectories on symmetric convex Hamiltonian energy surfacs, Discrete. Contin. Dyn. Syst., 32 (2012), 679-701.  doi: 10.3934/dcds.2012.32.679.  Google Scholar

[20]

D. Zhang, Maslov-type index and brake orbits in nonlinear Hamiltonian systems, Science in China Series A: Mathematics, 50 (2007), 761-772.  doi: 10.1007/s11425-007-0034-3.  Google Scholar

[21]

D. Zhang, Multiple symmetric brake orbits in bounded convex symmetric domains, Advanced. Nonlinear Studies, 6 (2006), 643-652.  doi: 10.1515/ans-2006-0408.  Google Scholar

show all references

References:
[1] A. Borel, Seminar on Transformation Groups, Princeton Univ. Press, Princeton, 1960.   Google Scholar
[2]

K. C. Chang, Infinite Dimensional Morse Theory and Multiple Solution Problems, Birkhauser, Basel, 1993. doi: 10.1007/978-1-4612-0385-8.  Google Scholar

[3]

H. Duan and H. Liu, Multiplicity and ellipticity of closed characteristics on compact star-shaped hypersurfaces in ${\Bbb R}^{2n}$, Calc. Var. Partial Differential Equations, 56 (2017), Art. 65, 30 pp, arXiv: 1601.03470. doi: 10.1007/s00526-017-1173-1.  Google Scholar

[4]

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

[5]

I. Ekeland and H. Hofer, Convex Hamiltonian energy surfaces and their periodic trajectories, Comm. Math. Phys, 113 (1987), 419-467.  doi: 10.1007/BF01221255.  Google Scholar

[6]

J. Gutt and J. Kang, On the minimal number of periodic orbits on some hypersurfaces in ${\Bbb R}^{2n}$, Ann. Inst. Fourier (Grenoble), 66 (2016), no. 6, 2485-2505. Google Scholar

[7]

C. Liu.-G., Maslov-type index theory for symplectic paths with Lagrangian boundary conditions, Adv. Nonlinear Stud, 7 (2007), 131-161.  doi: 10.1515/ans-2007-0107.  Google Scholar

[8]

J. Kang, Some remarks on symmetric periodic orbits in the restricted three-body problem, Discrete Contin. Dyn. Syst., 34 (2014), 5229-5245.  doi: 10.3934/dcds.2014.34.5229.  Google Scholar

[9]

C. Liu, Y. Long and C. Zhu, Multiplicity of closed characteristics on symmetric convex hypersurfaces in ${\Bbb R}^{2n}$, Math. Ann, 323 (2002), 201-215.  doi: 10.1007/s002089100257.  Google Scholar

[10]

C. Liu and D. Zhang, Seifert conjecture in the even convex case, Comm. Pure Appl. Math, 67 (2014), 1563-1604.  doi: 10.1002/cpa.21525.  Google Scholar

[11]

C. Liu and D. Zhang, Iteration theory of $L$-index and multiplicity of brake orbits, J. Differential Equations, 257 (2014), 1194–1245, arXiv: 0908.0021 [math.SG] doi: 10.1016/j.jde.2014.05.006.  Google Scholar

[12]

H. Liu, Y. Long and W. Wang, Resonance identities for closed characteristics on compact star-shaped hypersurfaces in ${\Bbb R}^2n$, J. Funct. Anal., 266 (2014), 5598-5638.  doi: 10.1016/j.jfa.2014.02.035.  Google Scholar

[13]

Y. Long, The topological structures of $\omega-subsets$ of symplectic groups, Acta Math., Sinica(English Series), 15 (1999), 255–268. doi: 10.1007/BF02650669.  Google Scholar

[14]

Y. Long, Index Theory for Symplectic Paths with Applictions, Progress in Mathematics. 2002. doi: 10.1007/978-3-0348-8175-3.  Google Scholar

[15]

Y. Long, Bott formula of the Maslov-type index theory, Pacific J. Math, 187 (1999), 113-149.  doi: 10.2140/pjm.1999.187.113.  Google Scholar

[16]

Y. Long, D. Zhang and C. Zhu, Multiple brake orbits in bounded convex symmetric domains, Adv.of Math, 203 (2006), 568-635.  doi: 10.1016/j.aim.2005.05.005.  Google Scholar

[17]

C. Viterbo, Equivariant Morse theory for starshaped Hamiltonian systems, Trans. Amer. Math.Soc, 311 (1989), 621-655.  doi: 10.1090/S0002-9947-1989-0978370-5.  Google Scholar

[18]

C. Viterbo, Une théorie de Morse pour les systèmes hamiltoniens étoilés, C. R. Acad. Sci Paris. Ser. I Math, 301 (1985), 487–489.  Google Scholar

[19]

W. Wang, Closed trajectories on symmetric convex Hamiltonian energy surfacs, Discrete. Contin. Dyn. Syst., 32 (2012), 679-701.  doi: 10.3934/dcds.2012.32.679.  Google Scholar

[20]

D. Zhang, Maslov-type index and brake orbits in nonlinear Hamiltonian systems, Science in China Series A: Mathematics, 50 (2007), 761-772.  doi: 10.1007/s11425-007-0034-3.  Google Scholar

[21]

D. Zhang, Multiple symmetric brake orbits in bounded convex symmetric domains, Advanced. Nonlinear Studies, 6 (2006), 643-652.  doi: 10.1515/ans-2006-0408.  Google Scholar

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