Julia sets and chaotic tunneling: II
Julia sets and chaotic tunneling: II
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DOI：
10.1088/1751-8113/42/26/265102
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发表时间：
2009-06
期刊：
影响因子：
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通讯作者：
A. Shudo;Y. Ishii;K. Ikeda
中科院分区：
文献类型：
--
作者：
A. Shudo;Y. Ishii;K. Ikeda
基于复杂的半经典理论来研究混沌隧穿，作为先前论文的延续（Shudo等人2009 J. Phys. A：Math. Theor. 42 265101）。本文研究了复杂经典轨道控制混沌隧穿的性质。结合数值研究的结果和严格的数学考虑的基础上理论的复杂动力系统导致我们的一个基本的数学定理，涉及到一组复杂的经典轨迹的隧道概率，称为Laputa链，和混沌组成部分的复杂相空间称为朱莉娅集。特别是，我们证明了隧道在不可积系统的机制是由一组密集的轨迹控制。这种机制是从根本上不同的可积系统中的一组稀疏的不变环面上的瞬子控制隧道。索赔的基本数学定理的物理意义进行了详细的数值研究。基于数值研究和Julia集的遍历性，我们提出了一个假设，保证了不可积系统中存在导致隧穿过程的复杂化轨道.这一假设支持了如下的混沌隧穿图景：隧穿轨迹在复空间中Julia集的稳定集和不稳定集的引导下通过真实的空间中的动力学势垒。
Chaotic tunneling is studied based on the complex semiclassical theory as a continuation of the previous paper (Shudo et al 2009 J. Phys. A: Math. Theor. 42 265101). In this paper, the nature of complex classical trajectories controlling chaotic tunneling is investigated. Combining the results of numerical investigations and rigorous mathematical considerations based on the theory of complex dynamical systems leads us to a fundamental mathematical theorem which relates the set of complex classical trajectories contributing to the tunneling probability, called the Laputa chains, and the chaotic component in the complexified phase space called the Julia set. In particular, we demonstrate that the mechanism for tunneling in non-integrable systems is controlled by a dense set of trajectories. This mechanism is radically different from the integrable system where a sparse set of instantons on invariant tori controls tunneling. The physical significance of claims in the fundamental mathematical theorem is numerically examined in detail. On the basis of the numerical studies and the ergodic nature of the Julia set, we finally propose a hypothesis which guarantees the existence of complexified trajectories contributing to the tunneling process in non-integrable systems. The hypothesis supports the following picture of chaotic tunneling: tunneling trajectories pass the dynamical barriers in the real space with the guidance of the stable and unstable sets of the Julia set in the complex space.