论文标题
单形图的通用可见性模式
Universal visibility patterns of unimodal maps
论文作者
论文摘要
我们通过研究可以从可见性映射推导的时间序列的几何结构来探索混乱中的顺序。该可见性映射将时间序列的每个点$（t，x（t））关联到其水平可见性盆地，即可以从每个点水平看到的时间序列的点数。我们将这项研究应用于产生同等分叉图的单峰图类图。我们使用经典的logistic图来说明本文的主要结果：分叉图的每个级联都有可见性模式，并在混乱开始时会收敛。 周期性时间序列的可见性模式以递归方式从可见性的基本块中产生。这种复发规则适用于所有周期性加倍级联。在特定窗口中，随着增长参数$ r $的变化，每个周期都会变化，这些块被复活地嵌入，以形成每个时期的可见性模式。在限制中，在混乱开始时，出现了无限的可见度模式，其中包含级联级联周期时间序列的所有可见性模式。 We have seen that these visibility patterns have specific properties: (i) the size of the elementary blocks depends on the period of the time series, (ii) certain time series sharing the same periodicity can have different elementary blocks and, therefore, different visibility patterns, (iii) since the 2-period and 3-period windows are unique, the respective elementary blocks, $ \{2\} $ and $ \{2 \, 3 \} $也是独特的，因此它们的可见性模式。我们探索其他低周期时间序列的可见性模式，并列举其每个周期性窗口的所有基本块。所有这些可见性模式都反映在混乱开始时相应的积累点，那里丢失了周期性。
We explore the order in chaos by studying the geometric structure of time series that can be deduced from a visibility mapping. This visibility mapping associates to each point $(t,x(t))$ of the time series to its horizontal visibility basin, i.e. the number of points of the time series that can be seen horizontally from each respective point. We apply this study to the class of unimodal maps that give rise to equivalent bifurcation diagrams. We use the classical logistic map to illustrate the main results of this paper: there are visibility patterns in each cascade of the bifurcation diagram, converging at the onset of chaos. The visibility pattern of a periodic time series is generated from elementary blocks of visibility in a recursive way. This rule of recurrence applies to all periodic-doubling cascades. Within a particular window, as the growth parameter $r$ varies and each period doubles, these blocks are recurrently embedded forming the visibility pattern for each period. In the limit, at the onset of chaos, an infinite pattern of visibility appears containing all visibility patterns of the periodic time series of the cascade. We have seen that these visibility patterns have specific properties: (i) the size of the elementary blocks depends on the period of the time series, (ii) certain time series sharing the same periodicity can have different elementary blocks and, therefore, different visibility patterns, (iii) since the 2-period and 3-period windows are unique, the respective elementary blocks, $ \{2\} $ and $ \{2 \, 3\}$, are also unique and thus, their visibility patterns. We explore the visibility patterns of other low-periodic time series and also enumerate all elementary blocks of each of their periodic windows. All of these visibility patterns are reflected in the corresponding accumulation points at the onset of chaos, where periodicity is lost.
