Ergodic and Kinetic Properties of Hamiltonian Dynamical Systems

为了阐明Hamilton动力系统的普遍规律，我们研究了复杂相空间结构所产生的轨迹的遍历性和动力学性质。对于各种哈密顿系统，相空间现象，如分裂的不变环面和周期轨迹的分叉进行了深入的研究，特别感兴趣的是它们对经典系统的轨迹的长期行为和量子系统的能级统计的影响。特别是，对于许多粒子系统，我们开发了一个复杂的现象，如集群运动，松弛过程和异常扩散的动力学理论。此外，我们还探索了将我们的理论结果应用于真实的实验中的可能性，主要结果如下：（1）混合哈密顿系统中的分形结构和相变现象揭示了环面与混沌过渡区的普适标度律。结果表明，许多粒子系统具有统计分布，如威布尔分布和对数威布尔分布，这是符合阿诺德扩散理论。（2）违反KAM条件的Hamilton系统的普适性质提出了一种系统研究违反KAM条件的Hamilton系统相空间的新方法。利用这种方法，我们成功地分析了重联现象，并准确地确定了全局混沌的临界阈值。3）经典相空间现象的量子特征我们建立了一个系统的方法来确定不可积台球系统能级统计的Berry-Robnik参数。在此基础上，我们证明了经典分岔现象对相应量子系统的能级统计性质有明显的影响。

In order to elucidate the universal laws of Hamiltonian dynamical systems, we studied ergodic and kinetic properties of trajectories generated by complex phase space structures. For various Hamiltonian systems, phase space phenomena such as breakup of invariant tori and bifurcations of periodic trajectories were thoroughly studied with particular interest in their effects on the long-term behavior of trajectories for classical systems and on the energy level statistics for quantum systems. In particular, for many particle systems, we developed a kinetic theory of complex phenomena such as clustering motions, relaxation processes and anomoulous diffusion. In addition, we pursued the possibility to apply our theoretical results to real experiments in the optical laser systems, where Hamiltonian dynamical theory can be applicable.Main results are summarized as follows :(1) Fractal structures and phase transition phenomena in mixed Hamiltonian systemsUniversal scaling laws are revealed in the transition regime between tori and chaos. It is shown that many particle systems exhibit statistical distributions such as Weibull distribution and Log-Weibull distribution, which are consistent with the Arnold diffusion theory.(2)Universal properties of Hamiltonian systems violating the KAM conditionA new method is proposed to systematically investigate the phase space of Hamiltonian systems violating the KAM condition. Using this method, we succeed in analyzing the reconnection phenomena and accurately determining the critical threshold for global chaos. It is shown that the set of critical threshold constitutes a fractal.3) Quantum signatures of classical phase space phenomenaWe establish a systematic method to determine the Berry-Robnik parameter of energy level statistics for non-integrable billiard systems. On the basis of this method, we demonstrate that the classical bifurcation phenomena clearly effect the level statistics properties of the corresponding quantum systems.