A collaborative investigation of a new bifurcation map — the "golden map" — whose periodic points encode the golden ratio in surprising geometric ways. Strong numerical evidence, no proof yet.

The Golden Map: why does φ keep appearing in its periodic points?

Numerically, the golden map's 2-cycle periodic points $\{p,q\}$ trace a segment of length $\sqrt{2}\,|p-q|$ that tends to $\varphi^2=1+\varphi=2.6180339\ldots$ as $r\to 2.4$. Can this be proven — solve the 2-cycle explicitly as a function of $r$ and show the limit is exactly $1+\varphi$? And is the fixed-point limit $1/\varphi$ (which holds exactly as $r\to\infty$) the underlying cause of the golden ratio appearing in the periodic-point geometry, or a separate coincidence?