Chaos Theory and Mathematics are directly connected with each other. Although people often think of Chaos theory as connected with weather events and scientific research, mathematicians are heavily involved in studying Chaos Theory as it applies to mathematical functions. In the Advanced Topics class, we use graphing calculator programs to explore the orbits of functions and analyze the different fates of the orbits, along with a search for any attractors that might exist for a given function.
Some definitions developed in class:
definition:function: a mathematical expression that takes an input and applies some rule to give an output
Ex: x2 - 1
definition:iterate: to use feedback (i.e. use the output as the next input)
definition:seed: a number used as a starting point in iterating a function
definition:fate of the orbit: a description of the ultimate fate of the sequence of numbers in the orbit (i.e. tends to infinity, fixed at 3, a 2-cycle, a 3-cycle, chaotic,...)
definition:attractor: if an infinite number of seeds all have the same fate for a given function, that fate is called an attractor
This introduction to chaos theory helps one see that simple functions like ".8x + 1" and "-4x2 + 4x" that appear very simple and determinstic, exhibit chaotic, unexpected behavior, when a recursive (feedback) process is utilized.
TRY THIS:
Pick a number, any number
Plug it in for x in .8x + 1
Take the result and plug that back in for x again in .8x + 1
Repeat this process
After some number of iterations, you should notice the list of numbers (the "orbit") converging on a particular number
Try this again with any other starting number ("seed"). Did you get the same result?
Try this same procedure.6x + 1
Same result?
If you explore many different functions, you will discover that some simple functions have attractors hidden in them, that some will lock into a 2-cycle (they oscillate back and forth between two values, infinitely), some will lock into a 3-cycle (or an n-cycle), some will take off to infinity, and some will lead to a seemingly chaotic and unpredictable list of numbers.
In the Advanced Topics class we also talk about the groundbreaking book by Edward Lorenz, called
The Essence of Chaos and the works of others, like Robert May, that brought together centuries of unexplored ideas whose time had come with the advent of the computer.
There are a myriad of physical systems that exhibit the ways in which chaos and order are interconnected...
A basic definition of a system that exhibits sensitive dependence would be one where a small change in the starting conditions of the system leads to a much larger change in the end result - ---for example, dropping a marble from anywhere on the edges of a hemispheric bowl would not exhibit SDIC, because no matter where you start, the marble will end up at the bottom in the center (this could be called an attractor).
Turning the bowl over, however, creates a system where a small change in where a marble is dropped on the top of the hemisphere does not result in only a change in where it lands, but in a very siginificant change (depending on the size of the bowl)
The Logistic Equation is an equation from population biology that, surprisingly, was discovered to exhibit chaotic behavior. It is a simple equation:
X=population between 0 (exinction) and 1 (carrying capacity), R = growth rate
Xnext year = RXthis year(1 - X)
This equation clearly uses feedback and when different rates are investigated, the fates of the populations have a wide variety of possible results...
In the class, we also explore the logistic equation and other functions using graphing calculator programs to experiment with and observe the "orbits", "seeds", and "fates of orbits" that lead to a variety of surpising results having connections with the question of the stability of the solar system , population biology, and many other areas. Here also, we see the connection with fractals , especially the Mandelbrot Set and Julia Sets. The bifurcation diagram (above) summarizes all the possible results of different growth rates in the logistic equation and is itself a fractal!
Zoom in on the bifurcation diagram (using the bifurcation applet), which summaizes all the information about the possible fates in population biology using the logistic equation. Look for the fractal in the chaos!
Newton's Clock - Chaos in the Solar System, by Ivars Peterson
A wonderful book that addresses the question of whether their is long-term stability in the solar system, which leads to a discussion about mathematical chaos and chaos in technology. The book is very well-written and gives a great historical overview of some ideas in astronomy, also.
The Essence of Chaos, by Edward Lorenz
Edward Lorenz is a meteorologist who broke new ground in looking at chaotic systems and how they impact the ability to predict weather.
Chaos - The Making of a New Science , by James Gleick
This was a seminal work that came out in the late 1980's that got many people turned on to Fractal Geometry and Chaos Theory. Gleick is a great author on scientific and mathematical topics.
Patterns of Chaos , by John Briggs
This is an oversized book that focuses mostly on images related to fractals and chaos theory in nature, science and art. The text adds some context, but doesn't stand on its own as a good overview of the topics. The images are quite striking.
