Skip to main content

\( \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Basic real and complex dynamicsA computational approach

Mark McClure

Front Matter

Preface

1Introduction

Surprise in Newton's method
The scope of chaos
Exercises

2The iteration of real functions

Basic notions
Graphical analysis
The classification of fixed points
Classification of periodic orbits
Parametrized families of functions
Conjugacy
The doubling map and chaos
A few notes on computation
Exercises

3Examples of complex iteration

Affine functions
\(z^2\text{:}\) The unit circle
\(z^2-2\text{:}\) A line segment
\(z^2-1\text{:}\) A fractal
A polynomial algorithm
\(z^2 - 0.123 + 0.745i\text{:}\) Douady's rabbit
A cubic with two attractive orbits
\(z+z^5\text{:}\) Neutrality
\(e^x/3\text{:}\) A hairy exponential
\(z/2 - 1/(2z)\text{:}\) Newton's method applied to a quadratic
\(z + \cot(z)\text{:}\) Newton's method applied to the cosine
Exercises

4Local theory of periodic orbits

Linearization near an attractive fixed point
Linearization near a repelling fixed point
Conjugation near a super-attractive fixed point
Neutral points
Infinity as a super-attractive fixed point
Exercises

5The quadratic family

Why focus on the quadratic family?
The quadratic escape criterion
The critical orbit
The Mandelbrot set
The components of the Mandelbrot set
Exercises

Authored in PreTeXt

Basic real and complex dynamicsA computational approach

Mark McClure
Department of Mathematics
University of North Carolina at Asheville
mcmcclur@unca.edu

October 26, 2017