论文标题
Fermi-Pasta-Ulam链的变化，调查
Variations on the Fermi-Pasta-Ulam chain, a survey
论文作者
论文摘要
我们将提出一项对低能量周期性的费米 - 帕斯塔 - 乌拉姆链的调查，并以“对称性的破坏”为主。经典的周期性FPU链（所有颗粒的相等质量）在2001年通过Rink分析，主要结论是，β链的正常形式始终是可集成的，并且在许多情况下，这也适用于Alfa链。带有交替质量的FPU链已经显示出某种对称性的破坏。可以鉴定出三个确切的定期解决方案家族，并与Chechin等人（1998-2005）在周期性解决方案的灌木丛中的结果有关。 2N颗粒的交替链作为链中的链条中存在，k 2n颗粒，k = 2，3，...。在大质量M的情况下，光学组和声学基团之间的相互作用已证明。 共鸣发挥的作用表明质量比的作用。对于任何数量的粒子和质量比，都不会产生1：1：1：...：1共振。一个有趣的案例是1：2：3的共振，在汉密尔顿 - 霍普夫妇分叉和破坏对称性混乱行为后产生的共振在希尔尼科夫·德瓦尼（Shilnikov-Devaney）的意义上。另一个有趣的情况是1：2：4共振。正如预期的那样，对各种情况的分析对复发现象有重大影响。这将通过数值结果来说明。
We will present a survey of low energy periodic Fermi-Pasta-Ulam chains with leading idea the "breaking of symmetry". The classical periodic FPU-chain (equal masses for all particles) was analysed by Rink in 2001 with main conclusions that the normal form of the beta-chain is always integrable and that in many cases this also holds for the alfa-chain. The FPU-chain with alternating masses already shows a certain breaking of symmetry. Three exact families of periodic solutions can be identified and a few exact invariant manifolds which are related to the results of Chechin et al.~(1998-2005) on bushes of periodic solutions. An alternating chain of 2n particles is present as submanifold in chains with k 2n particles, k=2, 3, ... . Interaction between the optical and acoustical group in the case of large mass m is demonstrated. The part played by resonance suggests the role of the mass ratios. The 1:1:1:...:1 resonance does not arise for any number of particles and mass ratios. An interesting case is the 1:2:3 resonance that produces after a Hamilton-Hopf bifurcation and breaking symmetry chaotic behaviour in the sense of Shilnikov-Devaney. Another interesting case is the 1:2:4 resonance. As expected the analysis of various cases has a significant impact on recurrence phenomena; this will be illustrated by numerical results.
