Periodically Driven anharmonic chain: Convergent Power Series and Numerics

We investigate the long time behavior of a pinned chain of $2N+1$ oscillators, indexed by $x\in \{-N,\dots ,N\}$. The system is subjected to an external driving force on the particle at $x=0$, of period $\theta =2\pi \mathrm{/}\omega$, 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 $\frac{{\omega }_0^2{q}_x^2}{2}+\frac{1}{2}(q_x-q_{x-1}{)}^2+\nu [V(q_x)+U(q_x-q_{x-1})]$, with $V^{\mathrm{\prime }\mathrm{\prime }}$ and $U^{\mathrm{\prime }\mathrm{\prime }}$ 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 $\mathrm{\mid }\nu \mathrm{\mid }<{\nu }_0$, yielding the long time periodic state of the system. Here ${\nu }_0$ depends only on the supremum norms of $V^{\mathrm{\prime }\mathrm{\prime }}$ and $U^{\mathrm{\prime }\mathrm{\prime }}$ 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.