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

Previous Article
Scaling properties of the Thue–Morse measure
DCDS Home
This Issue
Next Article
Equality of Kolmogorov-Sinai and permutation entropy for one-dimensional maps consisting of countably many monotone parts

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.
[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.
[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

[1]	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
[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]	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
[4]	Jian Zhai, Jianping Fang, Lanjun Li. Wave map with potential and hypersurface flow. Conference Publications, 2005, 2005 (Special) : 940-946. doi: 10.3934/proc.2005.2005.940
[5]	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
[6]	Salvatore A. Marano, Sunra J. N. Mosconi. Multiple solutions to elliptic inclusions via critical point theory on closed convex sets. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3087-3102. doi: 10.3934/dcds.2015.35.3087
[7]	Petr Kůrka, Vincent Penné, Sandro Vaienti. Dynamically defined recurrence dimension. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 137-146. doi: 10.3934/dcds.2002.8.137
[8]	Vittorio Martino. On the characteristic curvature operator. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1911-1922. doi: 10.3934/cpaa.2012.11.1911
[9]	Wan-Tong Li, Bin-Guo Wang. Attractor minimal sets for nonautonomous type-K competitive and semi-convex delay differential equations with applications. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 589-611. doi: 10.3934/dcds.2009.24.589
[10]	Jingang Xiong, Jiguang Bao. The obstacle problem for Monge-Ampère type equations in non-convex domains. Communications on Pure & Applied Analysis, 2011, 10 (1) : 59-68. doi: 10.3934/cpaa.2011.10.59
[11]	Jinkui Liu, Shengjie Li. Multivariate spectral DY-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2017, 13 (1) : 283-295. doi: 10.3934/jimo.2016017
[12]	Jia-Feng Liao, Yang Pu, Xiao-Feng Ke, Chun-Lei Tang. Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2157-2175. doi: 10.3934/cpaa.2017107
[13]	Luiz Felipe Nobili França. Partially hyperbolic sets with a dynamically minimal lamination. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2717-2729. doi: 10.3934/dcds.2018114
[14]	Rubén Caballero, Alexandre N. Carvalho, Pedro Marín-Rubio, José Valero. Robustness of dynamically gradient multivalued dynamical systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1049-1077. doi: 10.3934/dcdsb.2019006
[15]	Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[16]	Stefano Galatolo. Orbit complexity and data compression. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477
[17]	Shiqiu Liu, Frédérique Oggier. On applications of orbit codes to storage. Advances in Mathematics of Communications, 2016, 10 (1) : 113-130. doi: 10.3934/amc.2016.10.113
[18]	Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162
[19]	Sanjay K. Mazumdar, Cheng-Chew Lim. A neural network based anti-skid brake system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 321-338. doi: 10.3934/dcds.1999.5.321
[20]	B. Buffoni, F. Giannoni. Brake periodic orbits of prescribed Hamiltonian for indefinite Lagrangian systems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 217-222. doi: 10.3934/dcds.1995.1.217

American Institute of Mathematical Sciences