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

 is the textbook example of how a simple recurrence produces period-doubling and chaos. This paper introduces a different recurrence,

whose bifurcation diagram period-doubles considerably earlier than the logistic map and splits into two distinct self-similar structures. We call it the golden map. Extracting its periodic points and plotting them as coordinates reveals a recurring appearance of the golden ratio 

: the 2-cycle traces a segment of length 

, the 4- and 8-cycles form triangle configurations whose side ratios approach 

. The evidence is computational and a proof is open; we record the phenomena, the supporting data, and the questions a proof would have to answer.

 and its bifurcation diagram, I decided to look for a new bifurcation diagram with a different recurrence equation. The map studied here,

arose from that search. Generated in Mathematica, it period-doubles much sooner than the logistic map and, unexpectedly, produces two distinct self-similar structures rather than one. I call it the 

 map, because extracting its periodic points and plotting them as coordinates exposes a repeated — and surprising — relationship with the golden ratio.

Figure 1. The "golden" map. Period-doubling happens much sooner than in the logistic map, and the diagram splits into two distinct self-similar structures — an upper cascade and a lower one that hugs the axis.

For comparison, the classic logistic-map bifurcation diagram is shown below.

Figure 2. The logistic map bifurcation diagram, shown for comparison.

This document is an experimental report: it records the observations, the Mathematica procedures behind them, the supporting data, and the analysis I have so far. It does not contain a proof, and it is honest about that.

Everything below was computed in Mathematica. This section records exactly how the periodic points were found, how the bifurcation diagrams were generated, and how the points were plotted — so the results can be reproduced.

 I found the periodic points by iterating the recurrence at a fixed 

 and letting the orbit settle onto its cycle. The code below does this for 

a = 2;
tab = ConstantArray[{0, 0}, 2000];
Do[
  b = Abs[(1/Abs[a]) - ((1/Abs[a])^(1/1.1))] // N;
  tab[[i]] = {i, b};
  a = b;
  , {i, 1, 2000}]

ListPlot[tab]

Plotting the tab array shows the iterates snapping onto the periodic values almost immediately and holding there, so reading the cycle off the tail of the run is reliable. Replacing the exponent 

 gives the periodic points for any other 

 and keeping only the settled tail of each orbit produces Figure 1:

nestedlist =
  Table[{r, #} & /@
     NestList[Abs[(1/Abs[#]) - (1/Abs[#])^(1/r)] &, 0.01, 1000][[-100 ;;]],
   {r, Subdivide[1.1, 4, 1200]}];
flatlist = Flatten[nestedlist, 1];
ListPlot[flatlist]

 iterates, discarding the transient; and Subdivide[1.1, 4, 1200] sweeps 

 steps. The logistic diagram in Figure 2 is the same code with the map replaced by 

 To study a periodic window I take the cycle 

joined in order. Side lengths of the resulting figures come from the distance formula 

, and ratios of sides from similar-triangle relations. The next section carries this out for the 2-, 4-, and 8-cycles. Note that the golden ratio is never used anywhere in these computations — it only appears afterward, when the measured lengths and ratios are compared against it.

Figure 6. The 2-cycle for r = 1.1, plotted as the points (20.42, 0.015) and (0.015, 20.42). Its length is 28.86.

 to watch it as the 2-cycle begins merging into the 4-cycle:

   r          2-cycle length
  1.1         28.858162
  1.3          6.410408
  1.5          4.292866
  1.7          3.515384
  2.0          2.969218
  2.3          2.684565
  2.39         2.623744
  2.398        2.618725
  2.3991       2.618039
  2.39911009   2.618033   ->  1 + phi = 2.6180339887...

 the length decreases further as the 4-cycle points emerge). The golden ratio is already, subtly, present in this "golden" map.

. Plotted as coordinates the four points form a "two conjoined triangles" structure. For instance at 

Figure 8. The 4-cycle at r = 2.5, plotted as coordinates: two conjoined, similar triangles.

The two triangles are similar, so I labelled the six side lengths 

 increased. Figure 9 shows how the sides were labelled.

Figure 9. How the sides were labelled (shown at r = 3.18000001). Side lengths were found from the distance formula and similar-triangle relations.

    r           a/b          c/d
   2.5        1.969573     0.507724
   2.7        1.808199     0.553037
   2.9        1.706304     0.586062
   3.1        1.638973     0.610138
   3.15       1.625654     0.615137
   3.18       1.619270     0.617562
   3.18000001 1.617091     0.618394
   limit  ->  phi=1.618034  1/phi=0.618034

 the ratios move steadily toward the golden ratio 

. They hover near these values without ever landing exactly on them — an attraction, not an identity.

 the map has an 8-cycle, which plots the same way and produces a structure with two big triangles and two small ones. At 

Figure 11. The 8-cycle at r = 3.2678589, plotted the same way as the 4-cycle.

    r            a/b
   3.20        1.599735
   3.24        1.612915
   3.26        1.616762
   3.2678589   1.618034   ->  phi = 1.6180339887...

The same attraction to the golden ratio appears one level further down the cascade.

I tried to find a reason for the golden ratio appearing, so I considered the fixed-point condition 

and I wanted to see how the fixed point moves as 

 increases. I solved it in Mathematica, using a large exponent (here 

Solve[1/n - 1/Surd[n, 10000] == n, Reals]
N[%]
(*   n -> 0.618021   *)

 increases and I solve for the fixed point, 

 approaches the reciprocal of the golden ratio:

This limit is exact and elementary — the one place where 

 provably enters. The open question is whether it is the *cause* of the golden ratio appearing in the periodic-point geometry, or merely another place the same constant surfaces. The 2-cycle result (

, which is suggestive — but suggestive is not a proof, and the map's absolute values make the periodic-point algebra awkward to close in general.

The golden map ties a simple recurrence to the golden ratio through the geometry of its periodic points. The evidence is strong and reproducible, but every appearance of 

 here is currently empirical. A satisfying account would answer four questions.

. This is the most tractable target and the natural first result.

 For the 4- and 8-cycle triangle configurations, identify precisely which pairs of sides tend to 

, and prove the limit. Right now the ratios are measured, not derived.

 exactly. Determine whether this is the mechanism behind the 2-cycle and triangle appearances, or an independent coincidence.

 The two self-similar structures and the early period-doubling hint at a renormalization story. Is there a Feigenbaum-style fixed-point argument for this map, and does the golden ratio fall out of it?

If you have a background in dynamical systems, renormalization, or simply an eye for where 

 hides, question 1 is a concrete and inviting place to start. The project's bridge request asks exactly this.

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

Abstract

Introduction

Body

Analysis

Results

Proofs

Conclusion