Periodically Driven anharmonic chain: Convergent Power Series and Numerics

Abstract

We investigate the long time behavior of a pinned chain of $2N+1$ oscillators, indexed by $x \in\{-N,\ldots, N\}$. The system is subjected to an external driving force on the particle at $x=0$, of period $\theta=2\pi/\omega$, and to frictional damping $\gamma>0$ at both endpoints $x=-N$ and $N$. The oscillators interact with a pinned and nearest neighbor harmonic plus anharmonic potentials of the form $\frac{\omega_0^2 q_x^2}{2}+\frac12 (q_{x}-q_{x-1})^2 +\nu\left[V(q_x)+U(q_x-q_{x-1}) \right]$, with $V''$ and $U''$ bounded and $\nu\in \mathbb{R}$. We recall the recently proven convergence and the global stability of a perturbation series in powers of $\nu$ for $|\nu| < \nu_0$, yielding the long time periodic state of the system. Here $\nu_0$ depends only on the supremum norms of $V''$ and $U''$ and the distance of the set of non-negative integer multiplicities of $\omega$ from the interval $[\omega_0,\sqrt{\omega_0^2+4}]$ - the spectrum of the infinite harmonic chain for $\nu=0$. We describe also some numerical studies of this system going beyond our rigorous results.