Journal of Dynamical and Control Systems

, Volume 25, Issue 2, pp 159–174 | Cite as

Infinitely Many Rotating Periodic Solutions for Second-Order Hamiltonian Systems

  • Guanggang Liu
  • Yong LiEmail author
  • Xue Yang


In this paper, we consider a class of second-order Hamiltonian system in \(\mathbb {R}^{N}\) with combined nonlinearities. We will study the multiplicity of rotating periodic solutions, i.e., \(x(t+T)=Qx(t)\) with \(T>0\) and Q is an \(N\times N\) orthogonal matrix. In the case \(Q^{k}\neq I_{N}\) for any positive integer k, such a rotating periodic solution is just a quasi-periodic solution; In the case \(Q^{k}=I_{N}\) for some positive integer k, such a rotating periodic solution is just a subharmonic solution. We will use the Fountain Theorem and its dual form to obtain two sequences of rotating periodic solutions with the corresponding energy tending to infinity and zero respectively.


Second-order Hamiltonian systems Rotating periodic solutions Fountain theorem 

Mathematics Subject Classification 2010

34C25 37B30 37J45 



The authors are grateful to the reviewer for careful reading of the paper and helpful suggestions.


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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesLiaocheng UniversityLiaochengPeople’s Republic of China
  2. 2.School of Mathematics and Statistics and Center for Mathematics and Interdisciplinary SciencesNortheast Normal UniversityChangchunPeople’s Republic of China
  3. 3.College of MathematicsJilin UniversityChangchunPeople’s Republic of China

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