Uniformly travelling water waves from a dynamical systems viewpoint: some insights into bifurcations from Stokes’ family
Uniformly travelling water waves from a dynamical systems viewpoint: some insights into bifurcations from Stokes’ family
复制标题
从动力系统的角度来看均匀行进的水波：斯托克斯家族对分岔的一些见解
DOI：
--
复制
发表时间：
1992
影响因子：
3.7
通讯作者：
R. MacKay
中科院分区：
文献类型：
--
作者：
C. Baesens;R. MacKay
许多人对均匀行进的水波（无限深度的无粘性流体上的二维无旋重力波）的分岔进行的数值研究表明，均匀行进的水波具有可逆哈密顿公式，其中时间的作用由波系中的水平位置发挥作用。本文提出了这样的表述。基于这一观点，我们对斯托克斯周期波族的分岔给出了一些见解。数值证明，在振幅 A0 ≈ 0.40222 处存在一个“折叠点”。假设折叠非简并且存在相关的中心流形，这解释了为什么在 A0 的一侧出现一系列 p/q 分岔，其中 0 < p/q [les ] ½，按有理数顺序。其次，它解释了为什么在 A0 处没有观察到对称破缺分岔，这与一些人的预期相反。第三，它解释了为什么周期性均匀行波的分叉树看起来与保面积 Hénon 映射的分叉树非常相似。第四，它导致了对丰富多样的空间准周期波、异宿波和混沌波的预测。
Numerical work of many people on the bifurcations of uniformly travelling water waves (two-dimensional irrotational gravity waves on inviscid fluid of infinite depth) suggests that uniformly travelling water waves have a reversible Hamiltonian formulation, where the role of time is played by horizontal position in the wave frame. In this paper such a formulation is presented. Based on this viewpoint, some insights are given into bifurcations from Stokes’ family of periodic waves. It is demonstrated numerically that there is a ‘fold point’ at amplitude A0 ≈ 0.40222. Assuming non-degeneracy of the fold and existence of an associated centre manifold, this explains why a sequence of p/q-bifurcations occurs on one side of A0, with 0 < p/q [les ] ½, in the order of the rationals. Secondly, it explains why no symmetry-breaking bifurcation is observed at A0, contrary to the expectations of some. Thirdly, it explains why the bifurcation tree for periodic uniformly travelling waves looks so much like that for the area-preserving Hénon map. Fourthly, it leads to predictions of a rich variety of spatially quasi-periodic, heteroclinic and chaotic waves.