Normal Form of Periodic FPU Chain of Four Particles with Alternating Masses

The periodic Fermi-Pasta-Ulam (FPU) chain is a Hamiltonian system modeling a one-dimensional chain of oscillators with a periodic boundary condition and nearest-neighbor interactions. While previous studies typically restricted the potential function in the Hamiltonian to cubic or quartic terms, this work considers a more general setting. Our aim is to demonstrate how Birkhoff-Gustavson normalization, combined with symplectic transformations, can be systematically applied to study the dynamics of Hamiltonian systems. We focus on the periodic FPU chain of four particles with alternating masses . The presence of discrete symmetries in phase space simplifies the resulting normal form and reduces the system to one degree of freedom. The dynamics of this system depend on the parameter . We focus on two cases, and , which correspond to different classes of resonances. These cases exhibit topologically distinct phase space structures, which are classified in the final analysis. The results highlight the periodic FPU chain as a rich and tractable model for studying resonances and Hamiltonian dynamics in systems with symmetry and variable mass configurations.