That picture looks like this.
The population is predictable but it's not constant. It oscillates between two values.
We can keep going and find higher order patterns (e.g. 4-point cycles), but let's just make "r" really high (3.9) and see what happens.
We get no regularities. We get chaos. If you want to have endless fun playing with this equation, go
How does 4.669 figure in all this?