First-return maps as a unified renormalization scheme for dynamical systems.
First-return maps as a unified renormalization scheme for dynamical systems.
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首次返回映射作为动力系统的统一重整化方案。
DOI：
10.1103/physreva.35.1884
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发表时间：
1987
期刊：
影响因子：
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通讯作者：
Charles Tresser
中科院分区：
文献类型：
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作者：
I. Procaccia;Stefan Thomae;Charles Tresser
我们建议看看第一个返回映射到一个特定的相空间区域作为动力系统的统一重整化方案的基础。第一次返回的区域的选择由符号动态（例如，捏合序列）的相关轨迹。重整化群可以在符号水平上公式化，但是一旦转换成映射，它就产生了所述重整化方案。我们展示了如何充分研究的例子，通过周期加倍和准周期性的混沌发作适合这种方法，并认为这些问题实际上得到统一。统一也导致了一个概括，使我们能够研究的发病混乱的地图，属于更大的空间的功能比那些通常认为。在这些地图中，我们发现了许多新的混乱开始的场景。这些方案是物理相关的，因为考虑的地图是简单流的减少。我们提出了一个理论分析，这些新的情况下，并报告普遍的结果。最后，我们表明，所有可用的重正化群可以找到使用符号操作。
We propose to look at first-return maps into a specified region of phase space as a basis for a unified renormalization scheme for dynamical systems. The choice of the region for first return is dictated by the symbolic dynamics (e.g., kneading sequence) of the relevant trajectories. The renormalization group can be formulated on the symbolic level, but once translated to maps it yields the said renormalization scheme. We show how the well-studied examples of the onset of chaos via period doubling and quasiperiodicity fit into this approach, and argue that these problems get in fact unified. The unification leads also to a generalization that allows us to study the onset of chaos in maps that belong to larger spaces of functions than those usually considered. In these maps we discover a host of new scenarios for the onset of chaos. These scenarios are physically relevant since the maps considered are reductions of simple flows. We present a theoretical analysis of some of these new scenarios, and report universal results. Finally we show that all the available renormalization groups can be found using symbolic manipulations only.