Performing a Multi-Stage Mathematical and Informational Task
"lo-Fatou Petals, Siegel Disks, Herman Rings" as a Means of Developing Students' Creativity
Abstract
In this paper, using mathematical methods and computer experiments, we study the functions that form the periodic components of the Fatou set in holomorphic dynamics: Lo-Fatou petals, Siegel disks, and Herman rings. The study is based on the methodology of applying multi-stage mathematical-information tasks, which the authors used in their earlier publications. We continue to study polynomial dynamics and consider here the structures formed by parabolic fixed points. We study functions that have fixed neutral points. We develop algorithms and technologies for constructing Julia sets that have Lo-Fatou petals, Siegel disks, and Herman rings, not only in programming languages, but also in mathematical packages. Herman rings are formed when visualizing fractional-linear or hyperbolic mappings. If we trace the dynamics of the construction of such a ring, we obtain an irrational rotation of this ring. The listed structures of fixed points help to understand the behavior of the function near critical points and predict how the values of the function will change during multiple iterations. Studying the properties of the considered sets and their visualization when performing a multi-stage mathematical-information task contributes to the integration of mathematics and programming, provides an opportunity to develop flexibility, critical thinking, and intuition in students - the most important creative qualities.
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