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Stability of Subharmonic Solutions of First-Order Hamiltonian Systems with Anisotropic Growth

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Abstract

Using the dual Morse index theory, we study the stability of subharmonic solutions of first-order autonomous Hamiltonian systems with anisotropic growth, that is, we obtain a sequence of elliptic subharmonic solutions (that is, all its Floquet multipliers lying on the unit circle on the complex plane C).

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References

  1. Abbondandolo A. Morse Theory for Hamiltonian Systems. London: Chapman, Hall, 2001

    MATH  Google Scholar 

  2. An T, Wang Z. Periodic solutions of Hamiltonian systems with anisotropic growth. Commn Pure Appl Anal, 2010, 9: 1069–1082

    Article  MathSciNet  MATH  Google Scholar 

  3. Dell’Antonio G F, D’Onofrio B, Ekeland I. Stability from index estimates for periodic solutions of Lagrangian systems. J Differ Equ, 1994, 108: 170–189

    Article  MathSciNet  MATH  Google Scholar 

  4. D’Onofio B, Ekeland I. Hamiltonian systems with elliptic periodic orbits. Nonlinear Anal, 1990, 14: 11–21

    Article  MathSciNet  MATH  Google Scholar 

  5. Ekeland I. Convexity Methods in Hamiltonian Mechanics. Berlin: Springer, 1990

    Book  MATH  Google Scholar 

  6. Fan Z, Zhang D. Stability of the brake orbits on reversible symmetric compact convex hypersurfaces in R2n. J Fixed Point Theory Appl, 2017, 19: 503–527

    Article  MathSciNet  MATH  Google Scholar 

  7. Fei G. On periodic solutions of superquadratic Hamiltonian systems. Elect J Differ Equ, 2002, 8: 1–12

    Article  MathSciNet  MATH  Google Scholar 

  8. Giradi M, Matzeu M. Dual Morse index estimates for periodic solution of Hamiltonian systems in some nonconvex superquadratic case. Nonlinear Anal, 1991, 17: 481–497

    Article  MathSciNet  MATH  Google Scholar 

  9. Li C, Liu C G. The existence of nontrivial solutions of Hamiltonian systems with Lagrangian boundary conditions. Acta Math Sci, 2009, 29B(2): 313–326

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu C. The stability of subharmonic solutions for Hamiltonian systems. J Math Anal Appl, 1999, 240: 491–504

    Article  MathSciNet  MATH  Google Scholar 

  11. Liu C. The stability of the periodic solutions of second order Hamiltonian systems. Chinese Ann Math Ser B, 2000, 21: 225–232

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu C. Dual Morse index for subquadratic Hamiltonian systems and the stability of the closed characteristics (Chinese). Acta Math Sinica (Chin Ser), 2001, 44: 1073–1088

    MathSciNet  MATH  Google Scholar 

  13. Liu C, Long Y. Hyperbolic characteristics on starshaped surfaces. Ann Inst H Poincaré Anal Non linéaire, 1999, 16: 725–746

    Article  MathSciNet  MATH  Google Scholar 

  14. Liu C, Zhang Q. Nontrivial solutions for asymptotically linear Hamiltonian systems with Lagrangian boundary conditions. Acta Math Sci, 2012, 32B(4): 1545–1558

    MathSciNet  MATH  Google Scholar 

  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

    MathSciNet  MATH  Google Scholar 

  16. Long Y, Zhu C. Closed characteristics on compact convex hypersurfaces in R2n. Ann Math, 2002, 155: 317–368

    Article  MathSciNet  MATH  Google Scholar 

  17. 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

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Chungen Liu  (刘春根)

  2. School of Mathematics, Nankai University, Tianjin, 300071, China

    Xiaofei Zhang  (张晓飞)

Authors
  1. Chungen Liu  (刘春根)
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  2. Xiaofei Zhang  (张晓飞)
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Corresponding author

Correspondence to Chungen Liu  (刘春根).

Additional information

The first author was supported by NSFC (11471170, 11790271), innovation and development project of Guangzhou University.

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Liu, C., Zhang, X. Stability of Subharmonic Solutions of First-Order Hamiltonian Systems with Anisotropic Growth. Acta Math Sci 39, 111–118 (2019). https://doi.org/10.1007/s10473-019-0108-7

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  • DOI: https://doi.org/10.1007/s10473-019-0108-7

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