蛙跳涡对稳定性的半解析方法
Stability of Leapfrogging Vortex Pairs: A Semi-analytic Approach
摘要：
我们调查的稳定性的一个参数家庭的周期解的四涡问题称为“蛙跳”轨道。这些解由两对相同但符号相反的涡旋组成，W. Gröbli（1877）和A. E. H. Love（1883），并且可以通过与初始构型的几何形状相关的无量纲参数$α$来参数化。Acheson（2000）的模拟和Tophøj和Aref（2012）的数值Floquet分析都表明，当$1/α=ϕ^2$（其中$ϕ$是黄金分割率）时，会发生分叉。本研究旨在解释这一非凡价值的起源。使用一个技巧，从引力两体问题，我们改变变量，使Floquet问题在一个明确的形式，更适合分析。然后我们实现了G. W.希尔的谐波平衡法使用计算机代数构造一个快速收敛的序列的渐近逼近的分歧值，确认值发现较早。
英文摘要：
We investigate the stability of a one-parameter family of periodic solutions of the four-vortex problem known as `leapfrogging' orbits. These solutions, which consist of two pairs of identical yet oppositely-signed vortices, were known to W.\ Gröbli (1877) and A.\ E.\ H.\ Love (1883), and can be parameterized by a dimensionless parameter $α$ related to the geometry of the initial configuration. Simulations by Acheson (2000) and numerical Floquet analysis by Tophøj and Aref (2012) both indicate, to many digits, that the bifurcation occurs when $1/α=ϕ^2$, where $ϕ$ is the golden ratio. This study aims to explain the origin of this remarkable value. Using a trick from the gravitational two-body problem, we change variables to render the Floquet problem in an explicit form that is more amenable to analysis. We then implement G. W. Hill's method of harmonic balance to high order using computer algebra to construct a rapidly-converging sequence of asymptotic approximations to the bifurcation value, confirming the value found earlier.
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