.: Chaos and Order

The mathematics of chaos has been a fairly recent development in mathematics. The Schuelers have written a brief history of chaos theory that covers some of the highlights. I have written a first course in chaos, called Order - a closer look at chaos. It is available in two forms, both intended for senior high school/freshman college students and teachers. My recommendation is that you download the free program called Order, and install it on your computer. The Order program requires one of the 32 bit Windows operating systems. The installed size of the program on the hard drive is about 2 megabytes. Click on the OSetup link to download the installation program. Save it to a convenient location on your computer and run it to install Order. You may have to click an "Allow" button or equivalent if your security settings block the installation. If your computer is running Vista or Windows 7 you will need to download and install a Microsoft help reader to access the help files that come with Order. When you start Order, go to the "Open" menu item and open the "Getting Started" file. That will explain how to proceed.

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. The Java applets need an overhaul to improve some of the animation graphics, which I may get to in time. As they are they provide a lot of information. Just click on online Order to go to that course.

The following topics are covered in the Order program.

Introduction

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 requires one of the 32 bit Windows operating systems. The installed size of the program on the hard drive is about 2 megabytes. Click on the DLSetup link to download the installation program. Save it to a convenient location on your computer and run it to install DynaLab. You may have to click an "Allow" button or equivalent if your security settings block the installation. If your computer is running Vista or Windows 7 you will need to download and install a Microsoft help reader to access the help files that come with DynaLab. 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