not comfortable calling this one an attractor

What can happen (approximately) when you repeatedly do randomly either x ↦ z(x-1)³ or x ↦ z(x+1)³, for z=0.2*eⁱᵗ, starting with 0. I suspect that it doesn’t really matter where you start, in the sense that the set of things such a method converges to is always going to be a subset of what this one would be ideally. Maybe. Actually, that’s probably wrong. Perhaps. I don’t know. I can’t really tell. Somebody help me.