Title: Periodically Driven anharmonic chain: Convergent Power Series and Numerics Show Chinese title

Title: 周期驱动的非谐链：收敛幂级数和数值计算

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. Show Chinese abstract

Abstract: 我们研究了固定链的长期行为，该链由索引为$x \in\{-N,\ldots, N\}$的$2N+1$振子组成。系统在位置$x=0$的粒子上受到周期为$\theta=2\pi/\omega$的外部驱动力，并在两端点$x=-N$和$N$处受到摩擦阻尼$\gamma>0$。 振荡器与固定且最近邻的谐波加非谐波势相互作用，形式为$\frac{\omega_0^2 q_x^2}{2}+\frac12 (q_{x}-q_{x-1})^2 +\nu\left[V(q_x)+U(q_x-q_{x-1}) \right]$，其中$V''$和$U''$有界且$\nu\in \mathbb{R}$。我们回顾了最近证明的关于$\nu$的幂级数扰动展开的收敛性和全局稳定性，对于$|\nu| < \nu_0$，得到系统的长时间周期状态。 这里，$\nu_0$ 仅取决于 $V''$ 和 $U''$ 的上确界范数，以及 $\omega$ 的非负整数重数集与区间 $[\omega_0,\sqrt{\omega_0^2+4}]$（即 $\nu=0$ 时无限谐波链的谱）的距离。 我们还描述了该系统的一些数值研究，这些研究超越了我们严格的结果。