论文标题
用拉格朗日描述符揭示朱莉娅套装的分形结构
Unveiling the Fractal Structure of Julia Sets with Lagrangian Descriptors
论文作者
论文摘要
在本文中，我们通过拉格朗日描述符的方法探讨了朱莉娅（Julia）由复杂地图引起的，并分析了它们的基本动力学。特别是，我们查看了两个经典示例：二次映射$ z_ {n + 1} = z^2_n + c $，以及通过应用牛顿的方法生成的地图以找到复杂多项式的根源。为了实现这一目标，我们提供了这个标量诊断的扩展，该标量诊断能够揭示复杂平面中开放地图的相空间，从而使我们能够避免以越来越多的速率逃往无穷大的潜在问题。一个简单的想法是计算Lagrangian描述符的P-norm版本，而不是用于复杂平面上的点，而是用于扩展复杂平面的Riemann Sphere上的投影。我们用几个示例证明了该技术成功揭示了朱莉娅集合及其分形结构的丰富而复杂的动力学特征。
In this paper we explore by means of the method of Lagrangian descriptors the Julia sets arising from complex maps, and we analyze their underlying dynamics. In particular, we take a look at two classical examples: the quadratic mapping $z_{n+1} = z^2_n + c$, and the maps generated by applying Newton's method to find the roots of complex polynomials. To achieve this goal, we provide an extension of this scalar diagnostic that is capable of revealing the phase space of open maps in the complex plane, allowing us to avoid potential issues of orbits escaping to infinity at an increasing rate. The simple idea is to compute the p-norm version of Lagrangian descriptors, not for the points on the complex plane, but for their projections on the Riemann sphere in the extended complex plane. We demonstrate with several examples that this technique successfully reveals the rich and intricate dynamical features of Julia sets and their fractal structure.
