Chaos theory
 The work of a curious fellow

### .: Chaos and Order

The mathematics of chaos has been a fairly recent development in mathematics. I have written a first course in chaos, called Order - a closer look at chaos. It is intended for senior high school/freshman college students and teachers. It is available for download at no cost. The Order program has been tested in Windows XP, Win 7 32 bit and 64 bit and Win 8 32 bit and 64 bit. The installation program in which it is wrapped will match the installation to your operating system. The installed size of the program on the hard drive is about 2 megabytes. Before running the installation file, see the installation instructions for detailed suggestions on the steps to follow in installation. Click on the Order Install link and click on "Run". Depending on the security system settings you have, you will probably have to, in spite of perhaps multiple dire warnings, "Allow" the program to access your computer.

The other form of Order runs from the web in your browser. It depends heavily on Java applets so your equipment will need Java support, which comes installed on most computers.

Unfortunately the Oracle Corporation, which now manages the Java programming language, in 2014 implemented a security change that prevents applets that have been running without problems for decades from working. I have decided to leave the online version available because the text gives you a good idea of the contents of the downloadable program. You will not be able to use the dynamic illustrations until Oracle fixes this issue.

Just click on online Order to go to that course.

The following topics are covered in the Order program.

#### Introduction

Explains how to use the program and introduces the topics which follow:

#### Fundamentals

• Numbers, Functions and Graphs
• Extending Graphing Concepts
• Iteration and Attractors

#### Complexity in Simple Functions

• Phase Control Maps
• Exploring the Logistic Map
• Bifurcations
• Universality

#### Dynamical Systems

• Background
• Simple Pendulum
• Periodic Attractors
• Chaotic Attractors

#### Sets in the Complex Plane

• The Complex Plane
• The Mandelbrot Set
• Julia Sets

#### Generating Fractals

• Affine Transforms
• Multiple Affine Transforms

### .: DynaLab

DynaLab is a teaching and learning tool that stands alone as an introduction to the analysis of dynamical systems or may be used in conjunction with a course in that topic. Lessons authored by the program user may be illustrated by embedded dynamical system models of the user's design. The response of dynamical systems may be viewed as graphs vs. time, data tables, phase space projections, 3D phase space orbits, orbit sections including return maps, vector fields with nullclines and manifolds displayed or basins of attraction.

The program has been tested in Windows XP, Win 7 32 bit and 64 bit and Win 8 32 bit and 64 bit. The installation program in which it is wrapped will match the installation to your operating system. The installed size of the program on the hard drive is about 2 megabytes. Before running the installation file, see the installation instructions for detailed suggestions on the steps to follow in installation. Click on the DybaLab Install link and click on "Run". Depending on the security system settings you have, you will probably have to, in spite of perhaps multiple dire warnings, "Allow" the program to access your computer. When you start DynaLab, go to the "Open" menu item and open the "Getting Started" file. That will explain how to proceed. DynaLab is not available online. Java applets are not up to the job of handling the computational chores.

Data generated by the DynaLab models may be exported in tabular form for use in other programs. Also data in ASCII format may be imported. As part of the import algorithm, attractor reconstruction through delayed variables is available.

Many lessons are included with the program. These lessons were chosen to introduce some of the concepts used in the modern study of dynamical systems as well as to illustrate the capabilities of this program. A student who works through these examples in the order in which they are presented will be well positioned to succeed in a first course in dynamical systems taught at leading universities. Some of the lessons are listed below.

#### The Simple Harmonic Oscillator

• The simple harmonic oscillator is something that moves like a sine or cosine function.

#### The Simple Pendulum

• The simple pendulum is a pendulum with a rigid rod connecting one bob to one pivot, not necessarily one whose motion is simple.

#### The Duffing Mechanical Oscillator

• A nonlinear oscillator of the worst kind.

#### Systems in 1 Dimension

• Population Growth, Predation Without Feedback, First Order Phase Transition, Delayed Variable

#### Systems in 2 Dimensions

• Love Affairs, LRC Circuit, Predation With Feedback, Simple Harmonic Oscillator (again)

#### Systems in 3 Dimensions

• Folded Band, Lorenz System, Process Controls