Can we describe the connected components in the graph of $\cos\left(x\right)-\sin\left(x+y\right)=\cos\left(x^{2}y\right)$?

Question

The graph of $\cos\left(x\right)-\sin\left(x+y\right)=\cos\left(x^{2}y\right)$ is, perhaps unsurprisingly, pretty wild. Here is a Desmos version of it; a few screenshots are below, showing different scales of the graph. As can be seen, for small values of $x$ (say $-1.5 \le x \le 1.5$) the graph consists of individual simple closed loops; this pattern persists as you move up and down the $y$-axis, although the loops become smaller and more squished together.

However, for larger values of $x$, the loops merge together into what appears to be a single enormous zig-zag pattern. What appear in this image to be isolated points are actually an artifact of Desmos's inability to resolve the fine details; zooming in more closely reveals that they are actually connected. This raises the provocative question: how much of this function is actually connected?

My question is: are there any techniques that can be used to describe the connected components of this wild graph? For example, if everything with $x > 5$ path-connected?

Answer 1

Here is a plot in Mathematica ContourPlots with high resolution:

While each contour will be continuous (i.e., no isolated points as we see in the OP's plot), there may indeed be unconnected contour segments.