[euler_]

I really like the animation, but I'm confused by it. I thought that the whole thing with chaotic systems is that extremely slight differences in starting positions lead to very different outcomes. The continuity of the gif is surprising.

[aaachilless]

You'll see the chaotic behavior if you look around the vertical initial state at the top center of the graph. The initial states nearby evolve quite differently than each other.

"Chaotic" describes a system only locally around specific initial states. So a system can be chaotic near one initial state and stable near another.

[acedio]

Could be wrong, but I believe the areas you're referring to as continuous are the areas where it takes a while for the first flip to occur (if it ever does). Once a flip happens, neighboring cells seem to diverge quite quickly.