On infinite chains of periodically forced nonlinear oscillators. (English) Zbl 1299.34043

Pankov, Alexander

Differ. Integral Equ. 26, No. 7-8, 721-730 (2013).

This article proves the existence of time-periodic solutions for a lattice model of Frenkel-Kontorova chain type, \[ m_j\ddot x_{j}(t) = \alpha_{j}(x_{j+1}(t) - x_{j}(t)) - \alpha_{j-1}(x_{j}(t)- x_{j-1}(t)) + f_j(x_j(t)) - g_j(t), \] where \(x_j\) is the displacement of particle \(j\) in a chain, the nearest neighbour interaction is given by \(\alpha_j\), \(f_n\) is an on-site potential and \(g_n\) is a periodic force. The main result establishes the existence and uniqueness of a periodic odd solution under suitable assumptions. The proof relies on a reformulation as Dirichlet problem akin to Hamel’s treatment of the forced pendulum and a Banach fixed point argument.

Reviewer: Johannes Zimmer (Bath)

MSC:

34A33	Ordinary lattice differential equations
34A34	Nonlinear ordinary differential equations and systems
34C25	Periodic solutions to ordinary differential equations

Keywords:

lattice; Frenkel-Kontorova chain