Pseudo-Bifurcation diagram of f(x) = ln(x*x)

Here's a pseudo-bifurcation diagram for f(x) = ln(x2).  It came up in a discussion on the math-fun mailing list about repeatedly hitting the "ln(x)" and then the "x2" buttons on a calculator.  The original observation was how the values jumped around instead of settling in to a fixed final value or cycle of values.  This image helps to explain why.  It's not a real bifurcation diagram because the values jump around too much to give a nice, coherent picture.  Instead, I've plotted the actual function values for each of 10 iterations. The plot is symmetrical about x = 0 and red line is at x ~ -0.7035, the location of the fixed point.  The blue lines are at x ~ -1.2985 and x ~ 0.5224, the values in the period-2 cycle.  The plot resembles a series of onion layers, giving an indication of the repelling orbits of this function.