Feigenbaum吸引子的动力学
Dynamics towards the Feigenbaum attractor
摘要：
我们暴露在一个以前未知的细节水平的动态轨迹的功能，要么朝着费根鲍姆吸引或捕获其匹配的排斥。这些特征包括：i）吸引子和排斥子的原像集合彼此嵌入（密集）。ii）超循环吸引子族的吸引子和排斥子位置之间的分形边界的秩结构的极限形式得到原像布局。iii）每种情况下的联合原像集形成了吸引子或排斥子中定义明确的相空间间隙的无限多个族。iv）这些族中的每一个中的间隙可以根据幂律以减小的宽度排序，并且可以看到在由初始条件的均匀分布产生的动力学中顺序出现。v）对数周期调制的幂律与轨道接近吸引子（和排斥子）的速率相关，用间隙形成的进展来解释。（vi）阐明了原像结构的无穷层次性与吸引子的收敛速度律之间的关系。
英文摘要：
We expose at a previously unknown level of detail the features of the dynamics of trajectories that either evolve towards the Feigenbaum attractor or are captured by its matching repellor. Amongst these features are the following: i) The set of preimages of the attractor and of the repellor are embedded (dense) into each other. ii) The preimage layout is obtained as the limiting form of the rank structure of the fractal boundaries between attractor and repellor positions for the family of supercycle attractors. iii) The joint set of preimages for each case form an infinite number of families of well-defined phase-space gaps in the attractor or in the repellor. iv) The gaps in each of these families can be ordered with decreasing width in accord to power laws and are seen to appear sequentially in the dynamics generated by uniform distributions of initial conditions. v) The power law with log-periodic modulation associated to the rate of approach of trajectories towards the attractor (and to the repellor) is explained in terms of the progression of gap formation. vi) The relationship between the law of rate of convergence to the attractor and the inexhaustible hierarchy feature of the preimage structure is elucidated.
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