三自由度系统中跃迁准则与相空间结构
Transition criteria and phase space structures in a three degree of freedom system with dissipation
AI总结：
研究三自由度系统中通过指数为1的鞍点逃逸的动力学机制及相空间结构。
摘要：
通过指数为1的鞍点逃离势阱的现象在一些重要的物理系统中广泛存在。了解支配逃逸事件的准则和相空间几何结构在利用此类现象中起着重要作用，尤其是在存在现实摩擦或耗散力时。我们旨在将围绕鞍点的逃逸动力学研究从两个自由度扩展到三个自由度，提出了一种方法和相空间结构。既考虑了理想的守恒系统，也考虑了受扰动的耗散系统。我们定义了五维跃迁区域$\mathcal{T}_h$，作为给定初始能量$h$的初始条件集，这些轨迹会从鞍点的一侧逃逸到另一侧。不变流形论证表明，在六维相空间中，跃迁区域的边界$\partial \mathcal{T}_h$在守恒系统中拓扑上是一个四维超圆柱，在耗散系统中是一个四维超球面。跃迁区域$\mathcal{T}_h$可以通过三维椭球（三维圆柱）在三维配置空间中构建，其中每个点都有一个速度锥——导致跃迁的速度方向由锥给出，速度大小由初始能量和方向由两个给定限制的球面角度给出。为了说明我们的分析，我们考虑了一个具有两个势阱并通过指数为1的鞍点连接的示例系统。
英文摘要：
Escape from a potential well through an index-1 saddle can be widely found in some important physical systems. Knowing the criteria and phase space geometry that govern escape events plays an important role in making use of such phenomenon, particularly when realistic frictional or dissipative forces are present. We aim to extend the study the escape dynamics around the saddle from two degrees of freedom to three degrees of freedom, presenting both a methodology and phase space structures. Both the ideal conservative system and a perturbed, dissipative system are considered. We define the five-dimensional transition region, $\mathcal{T}_h$, as the set of initial conditions of a given initial energy $h$ for which the trajectories will escape from one side of the saddle to another. Invariant manifold arguments demonstrate that in the six-dimensional phase space, the boundary of the transition region, $\partial \mathcal{T}_h$, is topologically a four-dimensional hyper-cylinder in the conservative system, and a four-dimensional hyper-sphere in the dissipative system. The transition region $\mathcal{T}_h$ can be constructed by a solid three-dimensional ellipsoid (solid three-dimensional cylinder) in the three-dimensional configuration space, where at each point, there is a cone of velocity -- the velocity directions leading to transition are given by cones, with velocity magnitude given by the initial energy and the direction by two spherical angles with given limits. To illustrate our analysis, we consider an example system which has two potential minima connected by an index 1 saddle.
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