Finite length Fermi–Pasta–Ulam–Tsingou-type chains: Periodic forced dynamics and interaction inhomogeneity

We consider a finite length Fermi–Pasta–Ulam–Tsingou-type chain (FPUT) and determine conditions assuring that under periodic live loads there can be vibrations with a period T equal to the forcing one. Specifically, we prove existence of a connected set of T -periodic solutions that branch off the constant stationary one. In the analysis we consider general nonlinear interactions that include the standard choice for the FPUT α and β models but go beyond them. Indeed, we require only that the interaction between first neighbor mass points is merely a continuous function without any further specifications. We also consider the occurrence of inhomogeneities, induced by stiffening and/or weakening one or more springs with respect to the others; we thus prove under which conditions there exist periodic solutions with period equal to the loading one. Numerical simulations determine explicitly periodic solutions in special cases (7, 10, 21 and 51 vibrating mass points). They allow us to visualize and quantify in terms of internal (elastic) energy the effects of inhomogeneities on generic (thus not strictly periodic in space) chain dynamics.