Hamiltonian bifurcation perspective on two interacting vortex pairs: From symmetric to asymmetric leapfrogging, period doubling, and chaos
Hamiltonian bifurcation perspective on two interacting vortex pairs: From symmetric to asymmetric leapfrogging, period doubling, and chaos
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两个相互作用的涡对的哈密顿分岔视角：从对称到非对称蛙跳、周期倍增和混沌
DOI：
10.1103/physrevfluids.3.014401
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
V. Koukouloyannis
中科院分区：
文献类型：
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作者：
Brandon Whitchurch;P. Kevrekidis;V. Koukouloyannis
在这项工作中，我们研究了两个相互作用的涡对的动力学行为，每个涡对由两个在二维平面上相反循环的点涡组成。涡旋被视为有效粒子，它们之间的相互作用可以用经典力学来描述。为了得到系统相空间结构的图像，我们首先构造了一个典型的能量值的庞加莱截面。我们将相空间划分为对应于定性不同运动的不同区域，并在“真实”涡旋空间中演示了其不同的时间演化。我们的主要重点是跳跃周期轨道，围绕它我们确定一个区域，我们称之为“跳跃包络”，它主要涉及规则运动，如高阶周期解和准周期解。我们还识别了跳跃包络周围相平面的混沌区域以及所谓的漫步和编织运动。以能量为控制参数，构造了主跳越解及其不稳定性以及子分支不稳定性的分叉树。我们证明了跳跃解的对称性破缺不稳定性(与前人的工作一致)，并得到了相应的周期解的非对称分支。然后，我们描述了它们自己的不稳定性(包括倍周期的)和分叉，以努力提供一个更系统的视角来了解这个动力系统可用的运动类型。
In this work we study the dynamical behavior of two interacting vortex pairs, each one of them consisting of two point vortices with opposite circulation in the 2d plane. The vortices are considered as effective particles and their interaction can be desribed in classical mechanics terms. We first construct a Poincare section, for a typical value of the energy, in order to acquire a picture of the structure of the phase space of the system. We divide the phase space in different regions which correspond to qualitatively distinct motions and we demonstrate its different temporal evolution in the "real" vortex-space. Our main emphasis is on the leapfrogging periodic orbit, around which we identify a region that we term the "leapfrogging envelope" which involves mostly regular motions, such as higher order periodic and quasi-periodic solutions. We also identify the chaotic region of the phase plane surrounding the leapfrogging envelope as well as the so-called walkabout and braiding motions. Varying the energy as our control parameter, we construct a bifurcation tree of the main leapfrogging solution and its instabilities, as well as the instabilities of its daughter branches. We identify the symmetry-breaking instability of the leapfrogging solution (in line with earlier works), and also obtain the corresponding asymmetric branches of periodic solutions. We then characterize their own instabilities (including period doubling ones) and bifurcations in an effort to provide a more systematic perspective towards the types of motions available to this dynamical system.