Found a cool Non-autonomous Coupled System of ODEs

[TylerH]

By a fortuitous mistake in copy/pasting, I happened across this system. It exhibits very chaotic behavior for some initial values and spiral-like shapes for others. (About a 70%/30% split, respectively.)

$x'=cosy+sint$
$y'=sinx+cost$

Here's an album of it plotted from t=0 to 1000. The titles of the plot are x₀_y₀.
http://imgur.com/a/lbhrX
A lot of those have serious errors in their long term trends, but the overall type of pattern they take is, I believe, realistic. I believe so, because, as you can see in the video below, their vector fields are locally linear. From that, I intuitively draw the conclusion that there could be no serious numerical error, like jumping a phase line, but only small compounding error that only affects long term trend. That is to say, the only kind of numerical error present is the kind inherent to numerical methods, not anything specific to this ODE. (Correct me if my intuitive reasoning is incorrect.)

And, here's a video showing (x₀_y₀)=(1,1) with surrounding vector field from t=0 to 250:
https://vimeo.com/88323596

If anyone would like, I can post a video that is zoomed closer for the entire video, so as to see the local linearity better in the faster moving parts. I have only uploaded the one below so far because, IMO, it is more fun to watch that the one above.

 

[TylerH]

Also, I've reduced it to a system of very complex 2nd order noncoupled ODEs, but that doesn't seem to help in solving it. Mathematica can't solve it either. I'm almost convinced it isn't solvable. So it seems the only way to explore it is with numerical methods.