BRAKE ORBITS WITH MINIMAL PERIOD ESTIMATES OF FIRST-ORDER ANISOTROPIC HAMILTONIAN SYSTEMS

BRAKE ORBITS WITH MINIMAL PERIOD ESTIMATES OF FIRST-ORDER ANISOTROPIC HAMILTONIAN SYSTEMS

BRAKE ORBITS WITH MINIMAL PERIOD ESTIMATES OF FIRST-ORDER ANISOTROPIC HAMILTONIAN SYSTEMS

BRAKE ORBITS WITH MINIMAL PERIOD ESTIMATES OF FIRST-ORDER ANISOTROPIC HAMILTONIAN SYSTEMS

BRAKE ORBITS WITH MINIMAL PERIOD ESTIMATES OF FIRST-ORDER ANISOTROPIC HAMILTONIAN SYSTEMS

摘要/Abstract

参考文献

Metrics

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 10

Metrics

本文评价

推荐阅读 0

doi:10.1007/s10473-025-0205-8

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (2): 347-362.doi: 10.1007/s10473-025-0205-8

Xiaofei Zhang¹, Chungen Liu^2,*

1. School of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China;
2. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

收稿日期:2022-10-03 修回日期:2024-08-19 出版日期:2025-03-25 发布日期:2025-05-08

Xiaofei Zhang¹, Chungen Liu^2,*

1. School of Mathematics and Statistics, Shanxi Datong University, Datong 037009, China;
2. School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China

Received:2022-10-03 Revised:2024-08-19 Online:2025-03-25 Published:2025-05-08
Contact: *Chungen Liu, E-mail: liucg@nankai.edu.cn
About author:Xiaofei Zhang, E-mail: 835858094@qq.com
Supported by:
The first author's work was supported by the NSFC (12301138), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2021L377) and the Doctoral Scientific Research Foundation of Shanxi Datong University (2018-B-15). The second author's work was supported by the NSFC (12171108).

摘要： In this paper, the problem of brake orbits with minimal period estimates are considered for the first-order Hamiltonian systems with anisotropic growth, i.e., the Hamiltonian functions may have super-quadratic, sub-quadratic and quadratic behaviors simultaneously in different variable components.

关键词: Hamiltonian system, anisotropic growth, brake orbit, $L_{0}$ -index

Abstract: In this paper, the problem of brake orbits with minimal period estimates are considered for the first-order Hamiltonian systems with anisotropic growth, i.e., the Hamiltonian functions may have super-quadratic, sub-quadratic and quadratic behaviors simultaneously in different variable components.

Key words: Hamiltonian system, anisotropic growth, brake orbit, $L_{0}$ -index

中图分类号:

70H05

Xiaofei Zhang, Chungen Liu. BRAKE ORBITS WITH MINIMAL PERIOD ESTIMATES OF FIRST-ORDER ANISOTROPIC HAMILTONIAN SYSTEMS[J]. 数学物理学报(英文版), 2025, 45(2): 347-362.

Xiaofei Zhang, Chungen Liu. BRAKE ORBITS WITH MINIMAL PERIOD ESTIMATES OF FIRST-ORDER ANISOTROPIC HAMILTONIAN SYSTEMS[J]. Acta mathematica scientia,Series B, 2025, 45(2): 347-362.

[1] Abbondandolo A.Morse Theory for Hamiltonian Systems. London: Chapman and Hall, 2001
[2] Ambrosetti A, Benci V, Long Y. A note on the existence of multiple brake orbits. Nonlinear Anal, 1993, 21: 643-649
[3] An T, Wang Z. Periodic solutions of Hamiltonian systems with anisotropic growth. Comm Pure Appl Anal, 2010, 9: 1069-1082
[4] Duan H, Long Y, Zhu C. Index iteration theories for periodic orbits: Old and new. Nonlinear Anal, 2020, 201: 111999
[5] Fei G. On periodic solutions of superquadratic Hamiltonian systems. Electro J Differential Equations, 2002, 2002(8): 1-12
[6] Felmer P L. Periodic solutions of “super-quadratic” Hamiltonian systems. J Differential Equations, 1993, 18: 188-207
[7] Li C. The study of minimal period estimates for brake orbits of autonomous subquadratic Hamiltonian systems. Acta Math Sin Engl Ser, 2015, 31: 1645-1658
[8] Li C. Brake subharmonic solutions of subquadratic Hamiltonian systems. Chin Ann Math Ser B, 2016, 37: 405-418
[9] Li C, Liu C. Brake subharmonic solutions of first order Hamiltonian systems. Science China Mathematics, 2010, 53: 2719-2732
[10] Liu C. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete Contin Dyn Syst, 2010, 27: 337-355
[11] Liu C.Index Theory in Nonlinear Analysis. Beijing: Science Press, 2019
[12] Liu C, Long Y, Zhang D. Index iteration theory for brake orbit type solutions and applications. Analysis in Theory and Applications, 2021, 37: 129-156
[13] Liu C, Zhang D. Iteration theory of

L

$L$ -index and multiplicity of brake orbits. J Differential Equations, 2014, 257: 1194-1245
[14] Liu C, Zhang D. Seifert conjecture in the even convex case. Comm Pure Appl Math, 2014, 67: 1563-1604
[15] Liu C, Zhang X. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete Contin Dyn Syst, 2017, 37: 1559-1574
[16] Liu C, Zhang X. Stability of subharmonic solutions of first-order Hamiltonian systems with aisotropic growth. Acta Mathematica Scientia, 2019, 39B: 111-118
[17] Liu Z, Wang F, Zhang D. Brake orbits of a reversible even Hamiltonian system near an equilibrium. Acta Math Sin Engl Ser, 2022, 38: 263-280
[18] Long Y.Index Theory for Symplectic Paths with Applications. Berlin: Birkhauser, 2002
[19] Long Y, Zhang D, Zhu C. Multiple brake orbits in bounded convex symmetric domains. Advances in Math, 2006, 203: 568-635
[20] Xing Q, Guo F, Zhang X. One generalized critical point theorem and its applications on super-quadratic Hamiltonian systems. Taiwanese J Math, 2016, 20: 1093-1116
[21] Wang F, Zhang D. Multiple brake orbits of even Hamiltonian systems on torus. Science China Mathematics, 2020, 63: 1429-1440
[22] Xing J, Yang X, Li Y.Brake orbits of super-quadratic Hamiltonian systems. Proc Amer Math Soc, 2021, 149: 2179-2185
[23] Zhang D. Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems. Discrete Contin Dyn Syst, 2015, 35: 2227-2272
[24] Zhang X, Guo F. Existence of periodic solutions of a particular type of super-quadratic Hamiltonian systems. J Math Anal Appl, 2015, 421: 1587-1602
[25] Zhang X, Liu C. Brake orbits of first order convex Hamiltonian systems with particular anisotropic growth. Acta Math Sin Engl Ser, 2020, 36: 171-178
[26] Zhang X, Liu C, Lu X. Minimal periodic problem for brake orbits of first-order Hamiltonian systems. Topol Methods Nonlinear Anal, 2021, 57: 73-87
[27] Zhang X, Liu C. Minimal brake orbits of first-order convex Hamiltonian systems with anisotropic growth. Partial Differential Equations and Applications, 2021, 2: Art 45

Viewed

Full text

Abstract

Just accepted	Online first	Issue

0	0	38

From	Others	local

Times	37	1
Rate	97%	3%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed

Just accepted

Online first

Just accepted

Online first

[1]	Qingye Zhang, Chungen Liu. HOMOCLINIC SOLUTIONS NEAR THE ORIGIN FOR A CLASS OF FIRST ORDER HAMILTONIAN SYSTEMS^*[J]. 数学物理学报(英文版), 2023, 43(3): 1195-1210.
[2]	Hong Wang. HAMILTON-JACOBI EQUATIONS FOR A REGULAR CONTROLLED HAMILTONIAN SYSTEM AND ITS REDUCED SYSTEMS^*[J]. 数学物理学报(英文版), 2023, 43(2): 855-906.
[3]	王明伟, 郭飞. MULTIPLICITY OF PERIODIC SOLUTIONS FOR SECOND ORDER HAMILTONIAN SYSTEMS WITH MIXED NONLINEARITIES[J]. 数学物理学报(英文版), 2021, 41(2): 371-380.
[4]	刘春根, 张晓飞. STABILITY OF SUBHARMONIC SOLUTIONS OF FIRST-ORDER HAMILTONIAN SYSTEMS WITH ANISOTROPIC GROWTH[J]. 数学物理学报(英文版), 2019, 39(1): 111-118.
[5]	刘春根, 张清业. NONTRIVIAL SOLUTIONS FOR ASYMPTOTICALLY LINEAR HAMILTONIAN SYSTEMS WITH LAGRANGIAN BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2012, 32(4): 1545-1558.
[6]	万莉莉, 唐春雷. HOMOCLINIC ORBITS FOR A CLASS OF THE SECOND ORDER HAMILTONIAN SYSTEMS[J]. 数学物理学报(英文版), 2010, 30(1): 312-318.
[7]	李翀，刘春根. THE EXISTENCE OF NONTRIVIAL SOLUTIONS OF HAMILTONIAN SYSTEMS WITH LAGRANGIAN BOUNDARY CONDITIONS[J]. 数学物理学报(英文版), 2009, 29(2): 313-326.
[8]	丰建文, 陈士华. A CLASS OF QUADRATIC HAMILTONIAN SYSTEMS UNDER QUADRATIC PERTURBATION[J]. 数学物理学报(英文版), 2001, 21(2): 249-253.
[9]	赵晓华, 李继彬, 黄克累. HOMOCLINIC ORBITS IN PERTURBED GENERALIZED HAMILTONIAN SYSTEMS[J]. 数学物理学报(英文版), 1996, 16(4): 361-374.
[10]	赵晓华, 李继彬, 黄克累. PERIODIC ORBITS IN PERTURBED GENERALIZED HAMILTONIAN SYSTEMS[J]. 数学物理学报(英文版), 1995, 15(4): 370-384.