Convergent Power Series for Anharmonic Chain with Periodic Forcing

Abstract: We study the case of a pinned anharmonic chain of oscillators, with coordinates $$({\bf q},{\bf p})={(q_x, p_x):\,x\in\mathbb Z_N={-N, ......, N}}$$, subjected to an external driving force $F(\cdot)$ of period $\theta=2\pi/\omega$ acting on the oscillator at $x=0$. The system evolves according to a Hamiltonian dynamics with frictional damping, $\gamma&gt;0$, present at both endpoints $x=-N,N$. The Hamiltonian is given by $$H_N({\bf q},{\bf p})=\sum_{x\in\mathbb Z_N}\Big[\frac{p_x^2}2 + \frac12 (q_{x}-q_{x-1})² +\frac{\omega_0² q_x^{2}{2}+\nu\big(} V(q_x)+ U(q_x-q_{x-1})\big)\Big],$$ where $V(\cdot)$ and $U(\cdot)$ are $C<sup>2$ smooth anharmonic pinning and interaction potentials with bounded second derivatives, $\omega_0&gt;0$ and $\nu \in \mathbb R$. We prove that if no integer multiplicity of $\omega$ falls into the spectrum of the infinite harmonic system $I:=[\omega_0 ,\sqrt{\omega_0^2+4} ]$, then there exists a $\nu_0&gt;0$ such that for $\vert \nu\vert &lt;\nu_0$ the system approaches asymptotically in time a unique $\theta$-periodic solution, whose coordinates are given by a convergent power type series in $\nu$. The series coefficients are bounded and smooth functions of $\nu$. The value of $\nu_0$ is a lower bound on the radius of convergence, independent of $N$, the friction coefficient $\gamma&gt;0$ and ${\mathcal F}$. It depends only {on the supremum norm of $V&#39;&#39;(\cdot)$ and $U&#39;&#39;(\cdot)$} and the distance of the set of integer multiplicities of $\omega$ from $I$.