Update: I'm working on an interactive bifurcation diagram:
http://matt-diamond.com/sineMap.html
Here's the image when the starting coordinates are [0.5, 0.5]
The bifurcation diagrams differ depending on whether or not the coordinates are equal to each other or if one of the coordinates is 0 (essentially if the coordinate is on an axis or a diagonal). I personally think this "axial/diagonal" one looks cooler. :)
I was playing around with 2-dimensional recurrence relations in Grapher, and I stumbled upon this:
$$ x_{n+1} = \sin(k(y_n + x_n))\\ y_{n+1} = \sin(k(y_n - x_n)) $$ Here's an animation starting from an initial state of [0.1, 0.2] and k increasing from 0 to 2:
Same thing but an initial state of [1, 0]:
Here's some interesting behavior as k proceeds from 1 to 1.6:
I was wondering if this was a true chaotic map and if it might have any interesting properties.
Edit: Check this out... here's k = 1.4, $y_0 = 0$, $x_0$ increasing from 0 to 1:
It's like you can watch it shifting between two different limit cycles...
Edit 2: For those curious how I was able to create a recurrence relation in Grapher, here is a screenshot that demonstrates the main idea:
As far as I know, there's no hard-and-fast definition of a "true chaotic map". But there are certainly some that everybody agrees are chaotic - for example, the logistic map, with its famous bifurcation diagram.
For your map, I went ahead and generated a bifurcation diagram for the $x$-coordinates - specifically, in this graph, $k$ is along the horizontal axis, ranging from $0$ to $3$. For each value of $k$, I skipped the first $10000$ iterations, then plotted the next $10000$ $x_n$ along the vertical axis.
Those solid-black patches are chaos - even after $10000$ iterations, the $x$-coordinates were roughly evenly distributed across the interval $[0,1]$. But see those patches in the middle, where there are only a few points? Those are patches of stability, just like in the logistic map. That sort of behavior I'd call definitely chaotic.
One interesting difference from the logistic map is that it looks like it degenerates into chaos very quickly, with none or virtually none of the period-doubling characteristic of the logistic map. Or possibly the period-doubling is happening, but too fast to be seen on this scale.