论文标题
洛伦兹系统的分析研究：无限多个周期轨道的存在及其拓扑表征
Analytical study of The Lorenz system: existence of infinitely many periodic orbits and their topological characterization
论文作者
论文摘要
我们考虑洛伦兹方程，这是一个三维普通微分方程建模大气对流的系统。这些方程是混乱的，甚至很难在数字上研究，因此在七十年代引入了更简单的“几何模型”。动态系统中的经典问题之一是将原始方程与几何模型联系起来。塔克（Tucker）为经典参数值数字实现了这一点，并为一般值保持开放。在本文中，我们在分析上建立了与几何模型的关系，因为我们必须存在的一组参数值集。通过找到一种新的方法来应用用于研究表面动力学的拓扑工具到更复杂的三维流量的情况下，这可以促进这一点。
We consider the Lorenz equations, a system of three dimensional ordinary differential equations modeling atmospheric convection. These equations are chaotic and hard to study even numerically, and so a simpler "geometric model" has been introduced in the seventies. One of the classical problems in dynamical systems is to relate the original equations to the geometric model. This has been achieved numerically by Tucker for the classical parameter values, and remains open for general values. In this paper we establish analytically a relation to the geometric model, for a different set of parameter values that we prove must exist. This is facilitated by finding a novel way to apply topological tools developed for the study of surface dynamics to the more intricate case of three dimensional flows.
