Kvaternionska iteracija : delo diplomskega seminarja

Simčič, Luka (author), Prezelj, Jasna (mentor)

Glavni cilj diplomskega dela je opis in vizualizacija napolnjenih Juliajevih množic za kvaternionske kvadratne polinome kvaternionske spremenljivke, ki so posplošitev znanih napolnjenih Juliajevih množic za kompleksne kvadratne polinome kompleksne spremenljivke. Izpeljan je kriterij za neomejenost orbit ter opisane so napolnjene Juliajeve množice za določen razred kvadratnih polinomov. Njihov presek z dvema določenima ravninama sovpada z že znanima kompleksnima napolnjenima Juliajevima množicama, presek z ostalimi ravninami pa je s tem natanko določen. Priložena in opisana sta tudi koda za računanje z regularnimi funkcijami v Mathematici in iterativen algoritem za računanje orbit regularnih funkcij.

• Language: Slovenian
• Keywords: regularna funkcija, napolnjena Juliajeva množica, reprezentacijska formula
• Work type: Bachelor thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2023
• PID: 20.500.12556/RUL-149388
• UDC: 517.5
• COBISS.SI-ID: 163649027
• Publication date in RUL: 07.09.2023

Secondary language

• Language: English
• Title: Quaternionic Iteration

Abstract

The main goal of this work is the description and the visualization of filled Julia sets for quaternionic quadratic polynomials of a quaternionic variable which are a generalization of the known filled Julia sets for complex quadratic polynomials of a complex variable. A criterion is described for unboundedness of orbits and filled Julia sets of a certain class of polynomials are described. Their intersection with two specific planes coincides with the already known complex filled Julia sets and the intersection with the other planes is thereby precisely determined. Code for computing with regular functions in Mathematica is also attached and described, as well as an iterative algorithm for computing orbits of regular functions.

• Keywords: regular function, filled Julia set, representation formula