扰动三涡动力学
Perturbed Three Vortex Dynamics
摘要：
众所周知，在平面内理想流体中运动的三点涡的动力学可以用Hamilton形式表示，由此得到的运动方程在Liouville和Arnold意义下是完全可积的。本文研究的重点是平面上三涡系统的完全可积系统的正则性态（特别是周期运动）在某些（容许的）Hamilton扰动下的持续性。经过一个简短的调查的可积平面三涡系统的动力学，它表明，容许类扰动系统是足够广泛的，包括三个涡在半平面，三个同轴细长涡环在三个空间，和“限制”四涡动力学在一个平面。其中包括两类基本的可容许扰动结果：（i）用Kolmogorov-Arnold-Moser和Poincare-Birkhoff型参数证明不变环面和周期轨道的持久性的一般定理;和（ii）经典扰动理论性质的更具体和定量的结论，保证扰动系统的周期轨道的存在接近未扰动系统的周期，它们大量出现在中心附近。此外，提供了几个数值模拟，以说明定理的有效性，以及表明其局限性所表现出的过渡到混沌动力学。
英文摘要：
It is well known that the dynamics of three point vortices moving in an ideal fluid in the plane can be expressed in Hamiltonian form, where the resulting equations of motion are completely integrable in the sense of Liouville and Arnold. The focus of this investigation is on the persistence of regular behavior (especially periodic motion) associated to completely integrable systems for certain (admissible) kinds of Hamiltonian perturbations of the three vortex system in a plane. After a brief survey of the dynamics of the integrable planar three vortex system, it is shown that the admissible class of perturbed systems is broad enough to include three vortices in a half-plane, three coaxial slender vortex rings in three-space, and `restricted' four vortex dynamics in a plane. Included are two basic categories of results for admissible perturbations: (i) general theorems for the persistence of invariant tori and periodic orbits using Kolmogorov-Arnold-Moser and Poincare-Birkhoff type arguments; and (ii) more specific and quantitative conclusions of a classical perturbation theory nature guaranteeing the existence of periodic orbits of the perturbed system close to cycles of the unperturbed system, which occur in abundance near centers. In addition, several numerical simulations are provided to illustrate the validity of the theorems as well as indicating their limitations as manifested by transitions to chaotic dynamics.
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