论文标题
不变的托里（Tori
Nonexistence of Invariant Tori Transverse to Foliations: An Application of Converse KAM Theory
论文作者
论文摘要
不变流形对于对动态系统的定性理解至关重要。在这项工作中，我们探索和扩展了Mackay的Converse KAM条件，以获得足够的条件，即不存在不变表面的状态，这些表面横向至选择的一维叶面。我们展示了如何从系统的近似积分构建有用的叶子。该理论是针对两个模型的数值实现的，这是一个具有两波电位的粒子和Zaslavsky（Q-Flows）研究的Beltrami流。这些都是3D体积保存的流，它们分别说明了时间依赖性的汉密尔顿系统和不可压缩的流体中所见的动力学。通过数值和理论考虑，它揭示了如何选择以不同同源性捕获不变的Tori的叶面。
Invariant manifolds are of fundamental importance to the qualitative understanding of dynamical systems. In this work, we explore and extend MacKay's converse KAM condition to obtain a sufficient condition for the nonexistence of invariant surfaces that are transverse to a chosen 1D foliation. We show how useful foliations can be constructed from approximate integrals of the system. This theory is implemented numerically for two models, a particle in a two-wave potential and a Beltrami flow studied by Zaslavsky (Q-flows). These are both 3D volume-preserving flows, and they exemplify the dynamics seen in time-dependent Hamiltonian systems and incompressible fluids, respectively. Through both numerical and theoretical considerations, it is revealed how to choose foliations that capture the nonexistence of invariant tori with varying homologies.
