Renormalization of H´ enon Maps
Renormalization of H´ enon Maps
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H´enon 映射的重整化
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
M. Martens
中科院分区：
文献类型：
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作者：
M. Lyubich;M. Martens
在科学和物理实验中使用的许多模型中，在向混沌过渡时观察到倍周期级联。这些倍周期级联在一维动力学中得到了很好的理解。特别地，吸引子的微观几何性质不依赖于实际系统，它们是普适的。而且，两个不同映射的吸引子是光滑共轭的，它们是刚性的。强耗散Henon映射描述了系统的动力学的一部分，接近同宿切线，并经常在各种模型中观察到。对于这些地图的过渡到正熵也发生沿着倍周期级联。这些强耗散Henon映射可以看作是一维系统的扰动。事实上，一维系统的一些普遍几何性质存在于Henon映射中。然而，它们以一种更微妙的形式出现：在概率意义上，Henon吸引子的几何形状与它们的一维对应部分相同。这种现象被尊为概率的普遍性和刚性。
Period doubling cascades are observed at transition to chaos in many models used in the sciences and in physical experiments. These period doubling cascades are very well understood in one-dimensional dynamics. In particular, the microscopic geometrical properties of the attractors do not depend on the actual sys- tem, they are universal. Moreover, the attractors of two different maps are smoothly conjugate, they are rigid. Strongly dissipative Henon maps describe parts of the dynamics of systems close to a homoclinic tangency and are often observed in var- ious models. For these maps the transition to positive entropy also occurs along period doubling cascades. These strongly dissipative Henon maps can be consid- ered as perturbations of one-dimensional systems. Indeed, some of the universal geometrical properties of the one-dimensional systems are present in the Henon maps. However, they appear in a much more delicate form: in a probabilistic sense the geometry of the Henon attractors is the same as their one-dimensional counter part. This phenomenon is revered to as probabilistic universality and rigidity.