I built a bifurcation map from the recurrence 

. It period-doubles much sooner than the logistic map, so I have been calling it the golden map.

 from 1 up to about 2.4 the map settles into a 2-cycle, two values it bounces between. I plot those two values as the points 

 and measure the length of the line between them.

. The line keeps shrinking toward it and reaches it right around 

, which is exactly where the 2-cycle gives out and the 4-cycle begins. Past that point the length turns around and grows again.

So the golden ratio is sitting on the boundary between the 2-cycle and the 4-cycle. I have the numbers but not a reason.

In the golden map, the 2-cycle line length runs straight to 1 + φ

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