Homoclinic chaos in the dynamics of a general Bianchi type-IX model
Homoclinic chaos in the dynamics of a general Bianchi type-IX model
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一般 Bianchi IX 型模型动力学中的同宿混沌
DOI：
10.1103/physrevd.65.083511
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发表时间：
2002
影响因子：
5
通讯作者：
E. V. Tonini
中科院分区：
文献类型：
--
作者：
H. P. Oliveira;A. M. Almeida;I. Soares;E. V. Tonini
研究了具有三个尺度因子的一般Bianchi IX模型的动力学。该模型的物质含量被假定为移动的尘埃加上一个正的宇宙学常数。该模型在相空间有限区域内存在一个鞍-中心-中心型的临界点。这个临界点在相空间动力学中产生了稳定和不稳定四维管的拓扑$R \乘以S^3$，其中$R$是鞍形方向，$S^3$是中心-中心扇区中不稳定周期轨道的流形。动力学流动的一般特征是包含临界点和FRW奇点的动力学不变平面的轨道的振荡模式。我们证明了从靠近FRW奇点的临界点附近出现的一对管（一个稳定，一个不稳定）具有同斜横向交叉。同斜交流形具有拓扑$R \乘以S^2$，由同斜轨道组成，这些轨道对$S^3$中心流形是双渐近的。这是混沌在模型中的不变特征，在相空间中产生混沌集。该模型还在无穷远处呈现渐近DeSitter吸引子，初始条件集具有分形盆地边界，连接到逃逸到DeSitter构型（逃逸到膨胀），将临界点表征为混沌散射体。
The dynamics of a general Bianchi IX model with three scale factors is examined. The matter content of the model is assumed to be comoving dust plus a positive cosmological constant. The model presents a critical point of saddle-center-center type in the finite region of phase space. This critical point engenders in the phase space dynamics the topology of stable and unstable four dimensional tubes $R \times S^3$, where $R$ is a saddle direction and $S^3$ is the manifold of unstable periodic orbits in the center-center sector. A general characteristic of the dynamical flow is an oscillatory mode about orbits of an invariant plane of the dynamics which contains the critical point and a Friedmann-Robertson-Walker (FRW) singularity. We show that a pair of tubes (one stable, one unstable) emerging from the neighborhood of the critical point towards the FRW singularity have homoclinic transversal crossings. The homoclinic intersection manifold has topology $R \times S^2$ and is constituted of homoclinic orbits which are bi-asymptotic to the $S^3$ center-center manifold. This is an invariant signature of chaos in the model, and produces chaotic sets in phase space. The model also presents an asymptotic DeSitter attractor at infinity and initial conditions sets are shown to have fractal basin boundaries connected to the escape into the DeSitter configuration (escape into inflation), characterizing the critical point as a chaotic scatterer.