Unstable and exact periodic solutions of three-particles time-dependent FPU chains**Project supported by the National Natural Science Foundation of China (Grant Nos. 11301106, 11201288, and 11261013), the Natural Science Foundation of Guangxi Zhuang Autonomous Region, China (Grant No. 2014GXNSFBA118017), the Innovation Project of Graduate Education of Guangxi Zhuang Autonomous Region, China, (Grant No. YCSZ2014143), and the Guangxi Experiment Center of Information

Abstract

For lower dimensional Fermi–Pasta–Ulam (FPU) chains, the α-chain is completely integrable and the Hamiltonian of the β-chain can be identified with the Hénon–Heiles Hamiltonian. When the strengths α, β of the nonlinearities depend on time periodically with the same frequencies as the natural angular frequencies, the resonance phenomenon is inevitable. In this paper, for certain periodic functions α(t) and β(t) with resonance frequencies, we give the existence and stability of some nontrivial exact periodic solutions for a one-dimensional αβ-FPU model composed of three particles with periodic boundary conditions.