Interested in Chaos Theory, Complex Systems, Nonlinear Systems

[mathsciguy]

As the thread title says I'm interested in Chaos Theory, Complex Systems, and Nonlinear Systems. If I can help it, I'd like to study these at graduate level. My question is what kind and how much mathematics I'm supposed to know if I'm to study these?

 

[alan2]

What is your background? I know mathematicians who study this, physicists who study this, and mechanical engineers who study this. They all make significant contributions in different ways, some theoretical and some experimental. I think that one of the neatest things about dynamical systems is that it is very intuitive. You can read Strogatz to get very comfortable with the concepts and then read Wiggins from cover to cover. You can study from a purely mathematical point of view or you can go to the engineering school at Cornell and study dynamical systems, it's up to you. Start some reading.

 

[mathsciguy]

I'm still in my first year as a physics major so you can assume I don't have that much significant knowledge yet. I just asked so that I can prepare early if I actually decide to study this in the near future. I'd try looking up those you've mentioned, thanks.

 

[Number Nine]

At the very least, you'll want a strong background in differential equations and analysis.

 

[SophusLies]

I'm still in my first year as a physics major so you can assume I don't have that much significant knowledge yet. I just asked so that I can prepare early if I actually decide to study this in the near future. I'd try looking up those you've mentioned, thanks.

That's still pretty early to start thinking about nonlinear systems. Just learn as much as you can from your current courses and grab a differential equations book for the summer to self study. Any differential equations course or book will touch on nonlinear systems after you have a feeling of regular DE's. You're going to need some basic linear algebra to start also for the Jacobians. Analysis can't hurt either. I would highly recommend learning MATLAB right now as well. For me, nonlinear systems are best understood qualitatively because the math can get kinda nasty. Punching a non linear system through into a plot explains a whole lot.

 

[mathsciguy]

That's still pretty early to start thinking about nonlinear systems. Just learn as much as you can from your current courses and grab a differential equations book for the summer to self study. Any differential equations course or book will touch on nonlinear systems after you have a feeling of regular DE's. You're going to need some basic linear algebra to start also for the Jacobians. Analysis can't hurt either. I would highly recommend learning MATLAB right now as well. For me, nonlinear systems are best understood qualitatively because the math can get kinda nasty. Punching a non linear system through into a plot explains a whole lot.

Well yeah, it might appear to be a leap but I think having a goal is better. About differential equations, can it be done with only one semester knowledge of DE (I assume they only touch mostly ordinary differential equations)? Now about MATLAB, does it necessarily have to be it? Since I'm also learning python right now.

 

[bpatrick]

if you're looking to do that stuff in grad school, here's a short list of stuff i'd try to take while still an undergrad ... a lot of this is stuff you'll probably take anyway for your degree:

linear algebra
ODEs
classical mechanics
statistical mechanics

stuff you may not have to take for a physics degree but would be helpful for what you want / could count towards the degree depending on your school / department:

real analysis
complex analysis
graduate level classical mechanics class
graduate level linear algebra
dynamics / bifurcation theory
+any honors or 1st year grad classes related to chaos/nonlinear systems/etc... you can manage to fit in senior year.

 

[alan2]

Take all of the advice above. Go to math and take whatever 3rd and 4th year ode's they have. Take an undergrad analysis class. I would even recommend going to mechanical engineering if they offer a non-linear vibrations course. Buy Strogatz's book if you're new to this. Whether or not you understand it now depends on your background to date. You have lots of options, just don't scrimp on the math. I've always used MATLAB, I know nothing about python. Here is a fantastic toy that I've used for a long time, dfield/pplane, bookmark it:

http://math.rice.edu/~dfield/dfpp.html

 

[mathsciguy]

Thanks guys these just made everything clearer.