Skip to main content
SpringerLink
Log in

Multiple periodic solutions for non-canonical Hamiltonian systems with application to differential delay equations

  • Articles
  • Published:
Science China Mathematics Aims and scope Submit manuscript

Abstract

We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system. Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system is obtained by applying the Maslov index and Morse theory. As an application of the results, we study a class of non-autonomous differential delay equation which can be changed to non-canonical Hamiltonian system and obtain the existence of multiple periodic solutions for the equation by employing variational method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    MATH  Google Scholar 

  2. Amann H, Zehnder E. Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann Sc Norm Super Pisa Cl Sci, 1980, 7: 539–603

    MATH  MathSciNet  Google Scholar 

  3. Arnold V I. On a characteristics class entering into the quantization condition. Funct Anal Appl, 1967, 1: 1–13

    Article  Google Scholar 

  4. Chang K C. Critical Point Theory and Applications. Shanghai: Shanghai Science and Technology Press, 1986

    MATH  Google Scholar 

  5. Chang K C. Infinite Dimensional Morse Theory and Multiple Solution Problem. Boston: Birkhäuser, 1993

    Book  Google Scholar 

  6. Chang K C, Liu J, Liu M. Nntrivial periodic solutions for strong resonance Hamiltonian systems. Ann Inst H Poincaré Anal Non Linéaire, 1997, 14: 103–117

    Article  MATH  MathSciNet  Google Scholar 

  7. Cheng R. Symplectic transformations and a reduction method for asymptotically linear Hamiltonian systems. Acta Appl Math, 2010, 110: 209–214

    Article  MATH  MathSciNet  Google Scholar 

  8. Cheng R, Xu J. Periodic solutions for a class of non-autonomous differential delay equations. NoDEA Nonlinear Differential Equations Appl, 2009, 16: 793–809

    Article  MATH  MathSciNet  Google Scholar 

  9. Cheng R, Xu J, Zhang D. On the existence of periodic solutions for a class of non-autonomous differential delay equations. Topol Methods Nonlinear Anal, 2010, 35: 139–154

    MATH  MathSciNet  Google Scholar 

  10. Chow S N, Mallet-Paret J. The Fuller index and global Hopf bifurcation. J Differential Equations, 1978, 29: 66–85

    Article  MATH  MathSciNet  Google Scholar 

  11. Chow S N, Walther H O. Characteristic multipliers and stability of symmetric periodic solutions of \(\dot x = g(x(t - 1))\). Trans Amer Math Soc, 1988, 307: 127–142

    MATH  MathSciNet  Google Scholar 

  12. Cima A, Jibin L, Llibre J. Periodic solutions of delay differential equations with two delays via bi-Hamiltonian systems. Ann Differential Equations, 2001, 17: 2005–2014

    Google Scholar 

  13. Conley C, Zehnder E. Morse type index theory for flows and periodic solutions for Hamiltonian equations. Commun Pure Appl Math, 1984, 37: 207–253

    Article  MATH  MathSciNet  Google Scholar 

  14. Ding Y, Liu J. Periodic solutions of asymptotically linear Hamiltonian systems. J Systems Sci Math Sci, 1990, 9: 30–39

    MathSciNet  Google Scholar 

  15. Fei G. Maslov-type index and periodic solution of asymptotically linear Hamiltonian systems which are resonant at infinity. J Differential Equations, 1995, 121: 121–133

    Article  MATH  MathSciNet  Google Scholar 

  16. Fei G. Multiple periodic solutions of differential delay equations via Hamiltonian systems (I). Nonlinear Anal, 2006, 65: 25–39

    Article  MATH  MathSciNet  Google Scholar 

  17. Fei G, Qiu Q. Periodic solutions of asymptotically linear Hamiltonian systems. Chin Ann Math Ser B, 1997, 18: 359–372

    MATH  MathSciNet  Google Scholar 

  18. Guo Z, Yu J. Multiplicity results for periodic solutions to delay differential equations via critical point theory. J Differential Equations, 2005, 218: 15–35

    Article  MATH  MathSciNet  Google Scholar 

  19. Jekel S, Johnston C. A Hamiltonian with periodic orbits having several delays. J Differential Equations, 2006, 222: 425–438

    Article  MATH  MathSciNet  Google Scholar 

  20. Kaplan J, Yorke J. Ordinary differential equations which yield periodic solutions of differential delay equations. J Math Anal Appl, 1974, 48: 317–327

    Article  MATH  MathSciNet  Google Scholar 

  21. Li J, He X. Multiple periodic solutions of differential delay equations created by asymptotically linear Hamiltonian systems. Nonlinear Anal, 1998, 31: 45–54

    Article  MATH  MathSciNet  Google Scholar 

  22. Li J, He X, Liu Z. Hamiltonian symmetric groups and multiple periodic solutions of differential delay equations. Nonlinear Anal, 1999, 35: 457–474

    Article  MATH  MathSciNet  Google Scholar 

  23. Li S, Liu J. Morse theory and asymptotically linear Hamiltonian systems. J Differential Equations, 1989, 4: 53–73

    Article  Google Scholar 

  24. Liu C. Periodic solutions of asymptotically linear delay differential systems via Hamiltonian systems. J Differential Equations, 2012, 252: 5712–5734

    Article  MATH  MathSciNet  Google Scholar 

  25. Liu Z, Li J. Hamiltonian Systems and Periodic Solutions for Differential Delay Equations. Beijing: Science Publishhouse, 2000

    Google Scholar 

  26. Llibre J, Tarta A. Periodic solutions of delay equations with three delays via bi-Hamiltonian systems. Nonlinear Anal, 2006, 64: 2433–2441

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  28. Long Y. Index Theory for Symplectic Paths with Applications. Basel: Birkhäuser, 2002

    Book  MATH  Google Scholar 

  29. Long Y, Zehnder E. Morse theory for forced oscillations of asymptotically linear Hamiltonian systems. In: Stochastic Processes in Physics and Geometry. Singapore: World Scientific, 1990, 528–563

    Google Scholar 

  30. Maslov V. Theory of Perturbations and Asymptotic Methods. Paris: Ganthier-Villars, 1972

    Google Scholar 

  31. Mawhin J, Willem M. Critical Point Theory and Hamiltonian Systems. Berlin: Springer-Verlag, 1989

    Book  MATH  Google Scholar 

  32. Meyer K R, Hall G R. Introduction to Hamiltonian Dynamical Systems and the n-Body Problem. New York: Springer-Verlag, 1992

    Book  Google Scholar 

  33. Nussbaum R. Periodic solutions of special differential delay equations: An example in non-linear functional analysis. Proc Roy Soc Edinburgh Sect A, 1978, 81A: 131–151

    Article  MathSciNet  Google Scholar 

  34. Pisani L. Existence of periodic solutions for Hamiltonian systems at resonance. NoDEA Nonlinear Differential Equations Appl, 1997, 4: 61–76

    Article  MATH  MathSciNet  Google Scholar 

  35. Rabinowitz P H. Minimax Methods in Critical Point Theory with Application to Differential Equations. Providence, RI: Amer Math Soc, 1986

    Google Scholar 

  36. Su J. Existence of nontrivial periodic solutions for a class of resonance Hamiltonian systems. J Math Anal Appl, 1999, 233: 1–25

    Article  MATH  MathSciNet  Google Scholar 

  37. Su J. Nontrivial periodic solutions for the asymptotically linear Hamiltonian systems with resonance at infinity. J Differential Equations, 1998, 145: 252–273

    Article  MATH  MathSciNet  Google Scholar 

  38. Szulkin A. Cohomology and Morse theory for strongly indefinite functionals. Math Z, 1992, 209: 375–418

    Article  MATH  MathSciNet  Google Scholar 

  39. Viterbo C. A new obstruction to embedding Lagrangian tori. Invent Math, 1990, 100: 301–320

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics & Statistics, Nanjing University of Information Science & Technology, Nanjing, 210044, China

    Rong Cheng

  2. Department of Mathematics, Southeast University, Nanjing, 210096, China

    JunXiang Xu & FuBao Zhang

Authors
  1. Rong Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. JunXiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. FuBao Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rong Cheng.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cheng, R., Xu, J. & Zhang, F. Multiple periodic solutions for non-canonical Hamiltonian systems with application to differential delay equations. Sci. China Math. 57, 1625–1638 (2014). https://doi.org/10.1007/s11425-014-4829-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11425-014-4829-8

Keywords

MSC(2010)

Search

Navigation