Identifying this chaotic (?) recurrence relation

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:

• +1 nice observation! I would suggest a change of name for the control parameter $k$... you are following the "cinematic" evolution of the system through it, so it is more a kind of time $t$ control parameter. I am working also with discrete time systems using sine and cosine variables in the complex plane, and the results are quite amazing. This might give you ideas: math.stackexchange.com/questions/2654984/… – iadvd Feb 19 at 0:37
• sorry bad link in my former comment, this is the good one: math.stackexchange.com/q/2506900/189215 – iadvd Feb 19 at 3:02
• In general, you should identify the Fatou and Julia sets respectively.. points inside the Fatou set form normal families of iterates and converge to a fixed-point upon iteration.. maps in the Julia set diverge and are considered "chaotic" – crow Feb 20 at 18:01
• @iadvd thanks for the feedback... I'm not certain why I chose k... I was inspired by the Logistic Map which uses r, so perhaps I'll change it to that. I didn't really see it as a time property... I considered it just a factor in how the sequence is generated. – Matt D Feb 25 at 19:23
• @MattD How exactly did you create a recurrence relation in Grapher? – Frpzzd Feb 26 at 23:46

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.

• Wow, awesome! Thanks so much for this answer. When you created that bifurcation diagram, what did you use for the initial [x, y] coordinates? Also, what kind of software did you use to generate it? – Matt D Mar 2 at 21:51
• re: the rapid descent into chaos... if you look at this gif: i.stack.imgur.com/oJnln.gif you'll see that the spreading of x-coordinate values originally happens in an organized fashion, spiraling into a symmetrical arrangement of rings. Then those rings break down as the system descends into chaos. It would be interesting if one could produce a 3D bifurcation diagram that could capture this dynamic... – Matt D Mar 3 at 1:58
• The final image in my post (the one used to explain how I created recurrence relations in Grapher) contains an image of those symmetrical ring arrangements that arise at the boundary of chaos, btw... an interesting feature of this map. – Matt D Mar 3 at 2:05
• @MattD I used the same coordinates you did for the first one - (0.1, 0.2). As for software, I just wrote a quick Java program for it. A 3D version would be cool, but for that you'd need some sort of decent 3D-renderer. – Reese Mar 3 at 2:20
• I'm working on a 3D renderer right now... right now here's an interactive 2D view: matt-diamond.com/sineMap.html – Matt D Mar 13 at 23:23