Question: I'm a first year undergrad and I'm doing a small research extension on numerically solving ODEs. I have done the main ODE course at my university, as well as physics. The second part of the project constitutes using a method (linear multistep in my case) to solve a system of our choice and writing a report about it. I spoke to my supervisor and she said that a nonlinear example would be instructive since I am interested in plasma physics in the future. Also I'd like to do something involving fluids.

Problem is I have no idea where/how to find a specific example of a system modelled by a nonlinear ODE (I have found plenty of nonlinear PDE examples). So I was wondering if someone could tell me or guide me to an example, or how to find an example.

I am aware that this style of question may be frowned upon, but I've been looking for a while without success (and posted on other forums) so I thought I'd give it a shot.

Answer: 
The motion of an ordinary pendulum, i.e. a weight on a string, is an example of a non-linear ODE. You probably learned in school that it's linear, but this is only an approximation for small amplitudes.
Interested in the plasma physics, you can try to understand the [Debye-Hückel equation](http://en.wikipedia.org/wiki/Debye-H%C3%BCckel_equation). I'm not sure if the Wikipedia page is the best resource to start with. Most of the time people linearises this equation to find the Debye-length (this length has plenty *fathers*, but I do not remember neither who nor how many :-), but it may have some interesting things to do with the non-linear problem (or at least the higher order perturbed one). It's the problem of the density of screened free charges in a plasma.

This is an ODE only in 1D of course (note nevertheless the possibility to discuss the radial component when circular symmetry is supposed). I'm not sure if it's the more relevant one for plasma, but that's the only non-linear plasma or fluid example I know which is not a system of partial differential equations. 

Have fun.