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.
This simplest non-linear ODE problem and one that is rarely handled properly is one with step functions such as modeling friction. A simple spring mass system on a friction plane will provide the opportunity to try either

* Adjust numeric steps to discontinuity
* Plow over the discontinuity
* Adjust state directly or with iterations before next step is allowed
* Create a smooth function form of the discontinuity as an approximation

$$ \ddot{x} = -\omega^2 x -\mu m g \frac{ \dot{x} }{|\dot{x}|} $$

See example 6 [here](http://www.engineering.uiowa.edu/~sxiao/class/058-153/lecture-2.pdf) for a visual and a solution by inspection.