The damped Fermi–Pasta–Ulam oscillator

Abstract

The system \begin{equation*} \ddot{q}_k+\gamma \dot{q}_k=V'(q_{k+1}-q_k)-V'(q_k-q_{k-1})\qquad (k=1,\ldots,N-2) \end{equation*} is considered, where $00$ (fixed endpoints – this is the original Fermi–Pasta–Ulam oscillator provided that the damping coefficient $\gamma$ equals zero); $q_1(t)-q_0(t)= L/(N-1)$, $q_{N-1}(t)-q_{N-2}(t)= L/(N-1)$ (free endpoints); $q_0(t)=-(K-q_{N-2}(t))$, $q_{N-1}(t)=q_1(t)+K$, $K=\hbox{const.}$ (cycle). We prove that the unique equilibrium state of the system with fixed endpoints is asymptotically stable. We also prove that the system with free endpoints and the cycle asymptotically stop at an equilibrium state along their arbitrary motion, i.e., for every motion there is $q_1^\infty\in\mathbb{R}$ such that $\lim_{t\to\infty}q_k(t)=q_1^\infty+(k-1)\overline{r}$, $\lim_{t\to\infty}\dot q_k(t)=0$ $(k=1,\ldots,N-2)$, where the constant $\overline{r}$ is defined by the equation $V'(\overline{r})=0$.

Keywords

• asymptotic stability
• asymptotic stop
• invariance principle
• total mechanical energy