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May  2021, 41(5): 2095-2124. doi: 10.3934/dcds.2020354

Minimal period solutions in asymptotically linear Hamiltonian system with symmetries

Zhiping Fan 1, and  Duanzhi Zhang 2,,

1. 

School of Mathematical Sciences, Sun Yat-Sen University, Guangzhou 510300, China
2. 

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

* Corresponding author: Duanzhi Zhang

Received  November 2019 Revised  May 2020 Published  May 2021 Early access  October 2020

Fund Project: The first author is supported by the supported by the Fundamental Research Funds for the Central Universities (34000-31610273), Sun Yat-Sen University. The second author is supported by the NSF of China (17190271, 11422103, 11771341) and Nankai University

In this paper, applying the Maslov-type index theory for periodic orbits and brake orbits, we study the minimal period problems in asymptotically linear Hamiltonian systems with different symmetries. For the asymptotically linear semipositive even Hamiltonian systems, we prove that for any given $ T>0 $, there exists a central symmetric periodic solution with minimal period $ T $. Moreover, if the Hamiltonian systems are also reversible, we prove the existence of a central symmetric brake orbit with minimal period being either $ T $ or $ T/3 $. Also we give some other lower bound estimations for brake orbits case.

Keywords: Minimal period, Maslov-type indices, Brake orbits, Asymptotically linear Hamiltonian systems.
Mathematics Subject Classification: Primary: 37J12, 37J46; Secondary: 53D12.
Citation: Zhiping Fan, Duanzhi Zhang. Minimal period solutions in asymptotically linear Hamiltonian system with symmetries. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2095-2124. doi: 10.3934/dcds.2020354
References:
[1]

H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 7 (1980), 539-603.   Google Scholar

[2]

H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Manuscripta Math, 32 (1980), 149-189.  doi: 10.1007/BF01298187.  Google Scholar

[3]

A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann, 255 (1981), 405-421.  doi: 10.1007/BF01450713.  Google Scholar

[4]

S. E. Cappel, R. Lee and E. Y. Miller, On the maslov index, Comm. Pure Appl. Math, 47 (1994), 121-186.  doi: 10.1002/cpa.3160470202.  Google Scholar

[5]

K. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math, 34 (1981), 693-712.  doi: 10.1002/cpa.3160340503.  Google Scholar

[6]

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

[7]

K. Chang, J. Liu and M. Liu, Nontrivial periodic solutions for strong resonance Hamiltonian systems, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 14 (1997), 103–117. doi: 10.1016/S0294-1449(97)80150-3.  Google Scholar

[8]

F. Clark and I. Ekeland, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math, 33 (1980), 103-116.  doi: 10.1002/cpa.3160330202.  Google Scholar

[9]

D. Dong and Y. Long, The iteration theory of the Maslov-type index theory with applications to nonlinear Hamiltonian systems, Trans. Amer. Math. Soc, 349 (1997), 2619-2661.  doi: 10.1090/S0002-9947-97-01718-2.  Google Scholar

[10]

J. J. Duistermaat, On the Morse index in variational calculus, Adv. in Math, 21 (1976), 173-195.  doi: 10.1016/0001-8708(76)90074-8.  Google Scholar

[11]

I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems, Invent. Math, 81 (1985), 155-188.  doi: 10.1007/BF01388776.  Google Scholar

[12]

Z. Fan and D. Zhang, Multiple subharmonic solutions in Hamiltonian system with symmetries, submitted. Google Scholar

[13]

G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems, Nonlinear Anal, 27 (1996), 821-839.  doi: 10.1016/0362-546X(95)00077-9.  Google Scholar

[14]

G. Fei and Q. Qiu, Periodic solutions of asymptotically linear Hamiltonian systems, Chinese Ann. of Math. Ser. B, 18 (1997), 359-372.   Google Scholar

[15]

N. Ghoussoub, Location, multiplicity and Morse indices of min-max critical points, J Reine Angew Math, 417 (1991), 27-76.  doi: 10.1515/crll.1991.417.27.  Google Scholar

[16]

L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z, 219 (1995), 413-449.  doi: 10.1007/BF02572374.  Google Scholar

[17]

S. Li and J. Liu, Morse theory and asymptotic linear Hamiltonian system, J. Diff. Equ, 78 (1989), 53-73.  doi: 10.1016/0022-0396(89)90075-2.  Google Scholar

[18]

C. Liu, Asymptotically linear Hamiltonian systems with Lagrangian boundary conditions, Pacific J. Math, 232 (2007), 233-255.  doi: 10.2140/pjm.2007.232.233.  Google Scholar

[19]

C. Liu, 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

[20]

C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems, Discrete Contin. Dyn. Syst, 27 (2010), 337-355.  doi: 10.3934/dcds.2010.27.337.  Google Scholar

[21]

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

[22]

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

[23]

C. Liu and B. Zhou, Minimal $P$-symmetric period problem of first-order autonomous Hamiltonian systems, Front. Math. China, 12 (2017), 641-654.  doi: 10.1007/s11464-017-0627-2.  Google Scholar

[24]

Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems, Sci. China Ser. A, 33 (1990), 1409-1419.   Google Scholar

[25]

Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 10 (1993), 605–626. doi: 10.1016/S0294-1449(16)30199-8.  Google Scholar

[26]

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

[27]

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

[28]

J. Robin and D. Salamon, The Maslov index for paths, Topology, 32 (1993), 827-844.  doi: 10.1016/0040-9383(93)90052-W.  Google Scholar

[29]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 1986. doi: 10.1090/cbms/065.  Google Scholar

[30]

D. Zhang, Relative Morse index and multiple brake orbits of asymptotically linear Hamiltonian systems in the presence of symmetries, J. Differential Equations, 245 (2008), 925-938.  doi: 10.1016/j.jde.2008.04.020.  Google Scholar

[31]

D. Zhang, Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems, Sci. Chin. Math, 57 (2014), 81-96.  doi: 10.1007/s11425-013-4598-9.  Google Scholar

[32]

D. Zhang, Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems, Discrete Contin. Dyn. syst., 35 (2015), 2227-2272.  doi: 10.3934/dcds.2015.35.2227.  Google Scholar

show all references

References:
[1]

H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4), 7 (1980), 539-603.   Google Scholar

[2]

H. Amann and E. Zehnder, Periodic solutions of asymptotically linear Hamiltonian systems, Manuscripta Math, 32 (1980), 149-189.  doi: 10.1007/BF01298187.  Google Scholar

[3]

A. Ambrosetti and G. Mancini, Solutions of minimal period for a class of convex Hamiltonian systems, Math. Ann, 255 (1981), 405-421.  doi: 10.1007/BF01450713.  Google Scholar

[4]

S. E. Cappel, R. Lee and E. Y. Miller, On the maslov index, Comm. Pure Appl. Math, 47 (1994), 121-186.  doi: 10.1002/cpa.3160470202.  Google Scholar

[5]

K. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math, 34 (1981), 693-712.  doi: 10.1002/cpa.3160340503.  Google Scholar

[6]

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

[7]

K. Chang, J. Liu and M. Liu, Nontrivial periodic solutions for strong resonance Hamiltonian systems, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 14 (1997), 103–117. doi: 10.1016/S0294-1449(97)80150-3.  Google Scholar

[8]

F. Clark and I. Ekeland, Hamiltonian trajectories having prescribed minimal period, Comm. Pure Appl. Math, 33 (1980), 103-116.  doi: 10.1002/cpa.3160330202.  Google Scholar

[9]

D. Dong and Y. Long, The iteration theory of the Maslov-type index theory with applications to nonlinear Hamiltonian systems, Trans. Amer. Math. Soc, 349 (1997), 2619-2661.  doi: 10.1090/S0002-9947-97-01718-2.  Google Scholar

[10]

J. J. Duistermaat, On the Morse index in variational calculus, Adv. in Math, 21 (1976), 173-195.  doi: 10.1016/0001-8708(76)90074-8.  Google Scholar

[11]

I. Ekeland and H. Hofer, Periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems, Invent. Math, 81 (1985), 155-188.  doi: 10.1007/BF01388776.  Google Scholar

[12]

Z. Fan and D. Zhang, Multiple subharmonic solutions in Hamiltonian system with symmetries, submitted. Google Scholar

[13]

G. Fei and Q. Qiu, Minimal period solutions of nonlinear Hamiltonian systems, Nonlinear Anal, 27 (1996), 821-839.  doi: 10.1016/0362-546X(95)00077-9.  Google Scholar

[14]

G. Fei and Q. Qiu, Periodic solutions of asymptotically linear Hamiltonian systems, Chinese Ann. of Math. Ser. B, 18 (1997), 359-372.   Google Scholar

[15]

N. Ghoussoub, Location, multiplicity and Morse indices of min-max critical points, J Reine Angew Math, 417 (1991), 27-76.  doi: 10.1515/crll.1991.417.27.  Google Scholar

[16]

L. Hörmander, Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z, 219 (1995), 413-449.  doi: 10.1007/BF02572374.  Google Scholar

[17]

S. Li and J. Liu, Morse theory and asymptotic linear Hamiltonian system, J. Diff. Equ, 78 (1989), 53-73.  doi: 10.1016/0022-0396(89)90075-2.  Google Scholar

[18]

C. Liu, Asymptotically linear Hamiltonian systems with Lagrangian boundary conditions, Pacific J. Math, 232 (2007), 233-255.  doi: 10.2140/pjm.2007.232.233.  Google Scholar

[19]

C. Liu, 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

[20]

C. Liu, Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems, Discrete Contin. Dyn. Syst, 27 (2010), 337-355.  doi: 10.3934/dcds.2010.27.337.  Google Scholar

[21]

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

[22]

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

[23]

C. Liu and B. Zhou, Minimal $P$-symmetric period problem of first-order autonomous Hamiltonian systems, Front. Math. China, 12 (2017), 641-654.  doi: 10.1007/s11464-017-0627-2.  Google Scholar

[24]

Y. Long, Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems, Sci. China Ser. A, 33 (1990), 1409-1419.   Google Scholar

[25]

Y. Long, The minimal period problem of classical Hamiltonian systems with even potentials, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 10 (1993), 605–626. doi: 10.1016/S0294-1449(16)30199-8.  Google Scholar

[26]

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

[27]

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

[28]

J. Robin and D. Salamon, The Maslov index for paths, Topology, 32 (1993), 827-844.  doi: 10.1016/0040-9383(93)90052-W.  Google Scholar

[29]

P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, 1986. doi: 10.1090/cbms/065.  Google Scholar

[30]

D. Zhang, Relative Morse index and multiple brake orbits of asymptotically linear Hamiltonian systems in the presence of symmetries, J. Differential Equations, 245 (2008), 925-938.  doi: 10.1016/j.jde.2008.04.020.  Google Scholar

[31]

D. Zhang, Symmetric period solutions with prescribed minimal period for even autonomous semipositive Hamiltonian systems, Sci. Chin. Math, 57 (2014), 81-96.  doi: 10.1007/s11425-013-4598-9.  Google Scholar

[32]

D. Zhang, Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems, Discrete Contin. Dyn. syst., 35 (2015), 2227-2272.  doi: 10.3934/dcds.2015.35.2227.  Google Scholar

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