A periodic attractor repeats itself oscillating between values with each generation. A periodic attractor with period 2 oscillates between 2 values.
The state of the system is which periodic attractor the system is currently at.
A periodic attractor with period 4 oscillates between 4 values. As you increase the growthrate, R, the period of the system doubles. At a high enough growth rate (such as 4) there is no longer a fixed or periodic attractor and the system is chaotic. It displays sensitive dependence on initial conditions.
At growthrate, R, of 4, just a tiny change in the initial population % will have almost no effect in initial generations but the systems will diverge and follow radically different chaotic paths in later generations.
"Prediction becomes impossible." - Poincaré.
May in 1976 said the logistic map tells us that even with a simple model (the logistic map) with all parameters specified exactly, prediction is still impossible.
With a bifurcation diagram of R mapped to the X fixed point attractor we can see that as R reaches 3 we get to a period of 2, and as R approaches 4 we get to the onset of chaos at 3.569946 where there is no longer a period (it is infinite). At this point the system reaches the chaotic attractor, or strange attractor.
The logistic map displays the period doubling root to chaos.