We show interaction between high- and low-frequency modes in periodic $\alpha$-FPU chains with alternating large masses. This high-low frequency interaction is known if the number of particles $N$ is a 4-fold, our treatment discusses the difficult case where the number of particles $N=2p$ involves $p$ prime. A key role is played by identifying symmetric invariant manifolds, thus reducing the dimension of the problems drastically, and a Mathematica programme focused on these systems. We show explicitly high-low frequency interaction for systems with $2pn$ particles where $2\le p\le 47$ is prime and $n$ is an arbitrary natural number. In addition we have strong arguments for interactions in arbitrary large chains.

我们展示了周期性的高频和低频模式之间的相互作用 $\alpha$-具有交替大质量的 FPU 链。如果粒子数$N$ 是 4 倍，我们的处理讨论了粒子数 $N=2磷$ 涉及 $磷$主要的。识别对称不变流形发挥了关键作用，从而大大减少了问题的维度，以及专注于这些系统的Mathematica程序。我们明确地展示了系统的高低频交互$2磷n$ 粒子在哪里 $2\le 磷\le 47$ 是素数并且 $n$是一个任意的自然数。此外，我们对任意大型链中的交互有强有力的论据。