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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 and 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. 

[2]

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

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[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.

[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.

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. 

[2]

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

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[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.

[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.

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