Periodic Motions for FPU Lattice Systems with Asymptotically Quadratic Potentials

  • Jijiang Sun
  • Shiwang MaEmail author


In this paper, we consider the following autonomous Fermi–Pasta–Ulam lattice dynamical system:
$$\begin{aligned} \ddot{q}_{i}=\Phi '_{i-1}(q_{i-1}-q_{i})-\Phi '_{i}(q_{i}-q_{i+1}),\qquad i\in \mathbb {Z}, \end{aligned}$$
where \(\Phi _{i}\) denotes the interaction potential between two neighboring particles and \(q_{i}(t)\) is the state of the i-th particle. Supposing \(\Phi _{i}(x)\) is asymptotically quadratic at infinity, i.e., \(\Phi _{i}(x)\) tends to a quadratic function as \(|x|\rightarrow \infty \), for all \(T>0\), we obtain a nonzero T-periodic solution of finite energy. To our knowledge, there is no result dealing with this asymptotically quadratic case.


An infinite lattice of particles Asymptotically linear Variational method Periodic solutions 

Mathematics Subject Classification

Primary 37K60 Secondary 34C25 70F45 70G75 



The authors are very grateful to the anonymous referees for their valuable comments and helpful suggestions which have led to an improvement of the presentation of this paper. We also would like to sincerely thank Prof. Alexander Pankov for helpful discussions and suggestions.


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Authors and Affiliations

  1. 1.Department of MathematicsNanchang UniversityNanchangChina
  2. 2.School of Mathematical Sciences and LPMCNankai UniversityTianjinChina

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