Introduction to chaos theory

[Tricore]

Hello there

I have to write an interdisciplinary project at school, in mathematics and physics, where I have to go beyond what is covered in the syllabus for mathematics. I have chosen the subject of chaos theory.

Can you recommend any good introductions, on this subject? And would I need to study anything besides chaos theory at the same time, to understand it?

I don't know how my current level of education translates into the American system, but I am on my final year of taking an A-level in mathematics and physics, one year before I start university. I know about derivatives, and integrals and I am going to learn about differential equations and vector functions later this year. Can't figure out, if that means pre-calculus or not.

Thanks a lot in advance

 

[Mute]

How soon do you need to do your project? A lot of chaos theory depends very much on differential equations, such as the famous Lorenz attractor. A simpler route that might still be at your level and still shows chaos would be difference equations, which are sort of a discrete analog to differential equations. For example,

$$x_{n+1} = rx_n(1-x_n)$$

What this equation means is that you supply a value $x_0$, and it tells you what $x_1$. You plug $x_1$ back into the expression and it tells you what $x_2$ is, and so on. You can ask the question "What happens as we continue to iterate (keep plugging values of x back into) the equation?" It turns out that this depends on the parameter r. For 0 < r < 1, x_n -> 0, for 1 < r < 3 it convergences to (r-1)/r, then it starts oscillating betweens two values, then four values, then eight... eventually as r gets beyond about 3.57 the behaviour becomes chaotic and it never settles down to a fixed value. (This of course only works so long as you don't pick x_0 = 0 or 1!)

This equation is called the Logisitc map, and is a well known example of chaos.

Another thing you will want to look up when studying chaos theory are fractals, since the bifurcation diagram you get when you plot r versus the 'final' value of x_n for that map will exhibit fractal behaviour in the chaotic regime.

A book you might want to try to take a look at is "Nonlinear Dynamics and Chaos" by Strogatz. This will explain a lot of this.

 

[Tricore]

Thanks for your reply :-)

I have to write my project in week 49, so it's more than two months away, in fact, I am not even bound to my subject yet. I should be familiar with the subject, before week 49 though.

I've ordered the book on you recommended on the library.

How important would it be to know differential equations, in order to study chaos?

I know a bit about fractals already, as in I've seen the Mandelbrot and Julia set, along with a couple of others and heard about fractal dimensions, and know what complex numbers are. My adviser, has recommended that I stay away from fractals and focus on strange attractors though.

 

[Mute]

You still need to know a little about fractals if you're going to talk about strange attractors, because the set of points that consist of the strange attractor will have fractal properties.

As for knowing DEs, it depends on how in depth you want to go with the project, I guess. If you want to just be able to write a computer program to solve the DEs so that you can see the chaos, then maybe not knowing a lot is okay. However, if you wanted to do any "pen and paper" calculations yourself to get some basic results without using the computer, then you're going to need at least a basic understanding of differential equations, and a decent understanding if you wanted to do more complete analyses.

The important thing about chaos and ODEs is that in order to observe chaos in your (numerical) solutions you need a set of 3 coupled 1st order nonlinear differential equations (that is, 3 equations in 3 variables - "coupled" means that in the differential equation for one variable at least one of the other two variables appears, and "nonlinear" means that you have nonlinear functions of the variables in the expressions, for example powers of the variable or products of the different variables). Equivalently, you need one 3rd order nonlinear differential equation (3rd order means the highest derivative appearing is the 3rd derivative). It turns out that for only 2 coupled equations (or one 2nd order ODE) the topology of the solution space is sufficiently restricted such that you'll never see chaotic behaviour.

So, perhaps one simplification in your project is that you probably don't need to know how to solve ODEs in order to do it, since the equations that give you chaos aren't going to be solvable except by numerical methods. So, if you can get your understanding to the point where you can write a program to solve the equations, that may be good enough. One place where an understanding of solving ODEs would be important is determining the behaviour or solutions about "fixed points" - I'm not going to describe this here since I would have to drop a lot more terminology to explain it, but this is something you might want to look at when the book arrives.

 

[Tricore]

Alright, I think I have a fairly good idea on where to go from here. My book arrived today too, although I haven't really had the time to look at it yet.

Thanks a lot for your help, you really helped a bunch!