LAUSR

In this paper, we consider the following autonomous Fermi–Pasta–Ulam lattice dynamical system: $$\begin{aligned} \ddot{q}_{i}=\Phi '_{i-1}(q_{i-1}-q_{i})-\Phi '_{i}(q_{i}-q_{i+1}),\qquad i\in \mathbb {Z}, \end{aligned}$$q¨i=Φi-1′(qi-1-qi)-Φi′(qi-qi+1),i∈Z,where $$\Phi _{i}$$Φi denotes the interaction potential between two neighboring particles and $$q_{i}(t)$$qi(t) is the state of the i-th particle. Supposing $$\Phi _{i}(x)$$Φi(x) is asymptotically quadratic at infinity, i.e., $$\Phi _{i}(x)$$Φi(x) tends to a quadratic function as $$|x|\rightarrow \infty $$|x|→∞, for all $$T>0$$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.