Periodicity in Chaos

I would like to assume that there will be chaotic behavior in nearly every evolving system. I would also like to speculate that the Chaos is actually the search for the stable point of the whole dynamics of the process. Sometimes, there are apparently more than one stable fixed points and thus the system will keep jumping between them. The agents of stabilization will tend to set in to the stable fixed point while the dynamic behavior will try to make it move to next fixed point. During the violent competition between the competing processes, we observe that everything is pretty random. Hence may not look pretty nice.
After the system settles and revolves around the fixed point, the pattern looks pretty nice. As we can see it in the following picture (from Wikipedia):
(I know it makes no sense// I don't mind: why there would be any tendency of recurrence in the system?)