Step 1: pick an x (between 0 and 1)
Step 2: pick an r (between 1 and 4)
Step 3: Do the arithmetic
Step 4: Take the result, make it the new x, and do the arithmetic again.
Step 5: Rinse and repeat.
The graph for r = 2.7 and the beginning value of x = 0.5 is boring.
Look at this very simple equation.
Push r a bit higher (2.9), the x values take a bit longer to stabilize, but we still get the same boring pattern.
It gets more interesting if we set r = 3.3.