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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.02065 |
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Table of Contents:
- 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 $θ=2π/ω$, and to frictional damping $γ>0$ at both endpoints $x=-N$ and $N$. The oscillators interact with a pinned and nearest neighbor harmonic plus anharmonic potentials of the form $\frac{ω_0^2 q_x^2}{2}+\frac12 (q_{x}-q_{x-1})^2 +ν\left[V(q_x)+U(q_x-q_{x-1}) \right]$, with $V''$ and $U''$ bounded and $ν\in \mathbb{R}$. We recall the recently proven convergence and the global stability of a perturbation series in powers of $ν$ for $|ν| < ν_0$, yielding the long time periodic state of the system. Here $ν_0$ depends only on the supremum norms of $V''$ and $U''$ and the distance of the set of non-negative integer multiplicities of $ω$ from the interval $[ω_0,\sqrt{ω_0^2+4}]$ - the spectrum of the infinite harmonic chain for $ν=0$. We describe also some numerical studies of this system going beyond our rigorous results.