![The truncated tent map $T_h: [0, 1] \to [0, 1]$. Attributed to Matilde Marcolli - Sharkovsky’s Ordering and Chaos: Introduction to Fractal Geometry and Chaos.](https://prod-files-secure.s3.us-west-2.amazonaws.com/ac24fd92-2018-49be-80da-6ee154fc6a23/dc372c1d-719b-4292-99e7-f5db9b24ccdb/Untitled.png)
C. Hinsley
11 April 2024
This is a reference for bifurcation values of the truncated tent map associated with the attainment of each Sharkovsky type (including homoclinic bifurcations from the Sharkovsky stratification of $C^0(I, I)$).
The truncated tent map $T_h: [0, 1] \to [0, 1]$. Attributed to Matilde Marcolli - Sharkovsky’s Ordering and Chaos: Introduction to Fractal Geometry and Chaos.
| Sharkovsky-type bifurcation in $T_h$ | $h$ (exact) | $h$ (decimal expansion) |
|---|---|---|
| $F(1)$ | $0$ | $0$ |
| $F(2)$ | $2/3$ | $\sim 0.6666666667$ |
| $F(4)$ | $4/5$ | $0.8$ |
| $F(8)$ | $14/17$ | $\sim0.8235294118$ |
| $F(16)$ | $212/257$ | $\sim0.8249027237$ |
| $F(32)$ | $54062/65537$ | $\sim0.8249080672$ |
| $F(64)$ | $3542953172/4294967297$ | $\sim0.8249080673$ |
| … | … | … |
| $F(2^n), \quad n \gg 0$ | $\displaystyle 1-\frac{1}{2^{2^{n-1}}+1}\prod_{k=0}^{n-2} 2^{2^k}-1$ | $\sim0.8249080673$ |
| … | … | … |
| $H(2^\infty) = F(2^\infty)$ | $\displaystyle 1-\frac12\prod_{k=0}^\infty \left(1 - \frac{1}{2^{2^k}}\right)$ | $\sim0.8249080673$ |
| … | … | … |
| $H(2^n)=F(2^{n-1}(2\infty+1)), \quad n \gg 0$ | $\frac{(2^{2^n}-2)\left[\prod_{i=0}^{n-2}(2^{2^i}-1)\right]\left[1+\sum_{j=0}^{n-2}\frac{1}{\prod_{m=0}^j(2^{2^m}-1)}\right]+2^{2^{n-1}}+1}{2^{3\cdot2^{n-1}-1}+2^{2^n-1}}$ | $\sim0.8249080673$ |
| … | … | … |
| $H(64) = F(32(2\infty+1))$ | $\frac{32677975214510728853054477101}{39614081266355540833626750976}$ | $\sim0.8249080673$ |
| … | … | … |
| $H(32) = F(16(2\infty+1))$ | $\frac{116097260989651}{140739635838976}$ | $\sim0.8249080673$ |
| … | … | … |
| $F(16(2n+1)), \quad n \gg 0$ | $\frac{997252661111262790079278 \cdot 4294967296^{n-1}}{4294967295(2^{16(2n+1)}-1)} - \frac{3489660928}{4294967295(2^{16(2n+1)}-1)}?$ | $\sim0.8249080673$ |
| … | … | … |
| $F(80)$ | $\frac{997252661343453822397650}{1208925819614629174706175}$ | $\sim0.8249080673$ |
| $F(48)$ | $\frac{232190979026130}{281474976710655}$ | $\sim0.8249080673$ |
| $H(16) = F(8(2\infty+1))$ | $6946861/8421376$ | $\sim0.8249080673$ |
| … | … | … |
| $F(8(2n+1)), \quad n \gg 0$ | $\displaystyle \frac{3556792832 \cdot 65536^{n-1} - 212}{257(2^{8(2n+1)}-1)}$ | $\sim0.8249080673$ |
| $F(56)$ | $\frac{59440890630689580}{72057594037927935}$ | $\sim0.8249080673$ |
| $F(40)$ | $\frac{906996011820}{1099511627775}$ | $\sim0.8249080673$ |
| $F(24)$ | $13839660/16777215$ | $\sim0.8249080673$ |
| $H(8) = F(4(2\infty+1))$ | $1795/2176$ | $\sim0.8249080882$ |
| … | … | … |
| $F(4(2n+1)), \quad n \gg 0$ | $\displaystyle \frac{57440 \cdot 256^{n-1}-14}{17(2^{4(2n+1)}-1)}$ | $\sim0.8249080882$ |
| … | … | … |
| $F(20)$ | $288326/349525$ | $\sim0.8249080896$ |
| $F(12)$ | $1126/1365$ | $\sim0.8249084249$ |
| $H(4) = F(2(2\infty+1))$ | $33/40$ | $0.825$ |
| … | … | … |
| $F(2(2n+1)), \quad n \gg 0$ | $\displaystyle \frac{33 \cdot 16^n-24}{10(2^{2(2n+1)}-1)}$ | $\sim0.825$ |
| … | … | … |
| $F(26)$ | $55364812/67108863$ | $\sim0.8250000004$ |
| $F(22)$ | $3460300/4194303$ | $\sim0.8250000060$ |
| $F(18)$ | $216268/262143$ | $\sim0.8250000954$ |
| $F(14)$ | $13516/16383$ | $\sim0.8250015260$ |
| $F(10)$ | $844/1023$ | $\sim0.8250244379$ |
| $F(6)$ | $52/63$ | $\sim0.8253968254$ |
| $H(2) = F(2\infty+1)$ | $5/6$ | $\sim0.8333333333$ |
| … | … | … |
| $F(2n+1), \quad n \gg 0$ | $\displaystyle \frac{5 \cdot 4^n-2}{3(2^{2n+1}-1)}$ | $\sim0.8333333333$ |
| … | … | … |
| $F(31)$ | $1789569706/2147483647$ | $\sim0.8333333334$ |
| $F(29)$ | $447392426/536870911$ | $\sim0.8333333336$ |
| $F(27)$ | $111848106/134217727$ | $\sim0.8333333346$ |
| $F(25)$ | $27962026/33554431$ | $\sim0.8333333383$ |
| $F(23)$ | $6990506/8388607$ | $\sim0.8333333532$ |
| $F(21)$ | $1747626/2097151$ | $\sim0.8333334128$ |
| $F(19)$ | $436906/524287$ | $\sim0.8333336512$ |
| $F(17)$ | $109226/131071$ | $\sim0.8333346049$ |
| $F(15)$ | $27306/32767$ | $\sim0.8333384198$ |
| $F(13)$ | $6826/8191$ | $\sim0.8333536809$ |
| $F(11)$ | $1706/2047$ | $\sim0.8334147533$ |
| $F(9)$ | $426/511$ | $\sim0.8336594912$ |
| $F(7)$ | $106/127$ | $\sim0.8346456693$ |
| $F(5)$ | $26/31$ | $\sim0.8387096774$ |
| $F(3)$ | $6/7$ | $\sim0.8571428571$ |
| $H(1)$ | $1$ | $1$ |
The bifurcation value to $F(2^\infty)$ is twice the Prouhet-Thue-Morse constant. I obtained the closed form of this value from this Wikipedia page. This value is transcendental.
Notably, the truncated tent map has a quadratic limit in its sequence of period-doubling bifurcation values, rather than the geometric limit observed for quadratic maps like the logistic map (or any smooth unimodal map with single turning point).
The denominators for $F(2^k(2n+1))$ are easy: they are $2^{2^k(2n+1)}-1$. The numerators are governed by affine recurrence relations.
For the homoclinic bifurcation values, the denominators are given by $2^{3\cdot2^{n-1}-1}+2^{2\cdot2^{n-1}-1}$ for $H(2^n), n \geq 1$. This is a multiple of Fermat numbers, also appearing in an article by Bugeaud and Queffélec, Theorem 1. Meanwhile, the numerators are given by $k_n(2^{2^n}-2)+2^{2^{n-1}}+1$, where $k_1 = 1$ and $k_{n+1} = k_n(2^{2^{n-1}}-1)+1$ (OEIS A162634), a recurrence relation which has closed form
$$ k_n = \left[\prod_{i=0}^{n-2}(2^{2^i}-1)\right]\left[1+\sum_{j=0}^{n-2}\frac{1}{\prod_{m=0}^j(2^{2^m}-1)}\right]. $$
I.e., the bifurcation value for $H(2^n)$ is
$$ h = \frac{(2^{2^n}-2)\left[\prod_{i=0}^{n-2}(2^{2^i}-1)\right]\left[1+\sum_{j=0}^{n-2}\frac{1}{\prod_{m=0}^j(2^{2^m}-1)}\right]+2^{2^{n-1}}+1}{2^{3\cdot2^{n-1}-1}+2^{2^n-1}}. $$
The code for the first cell in CobwebTool to construct the truncated tent map is as follows:
from typing import List, Tuple
h = 1
orbits : List[List[float]] = [[]]
extra_points : List[Tuple[float, float]] = [(0,0),(1,0)] + ([(0.5,1)] if h==1 else [] if h==0 else [(h/2,h), (1-h/2,h)])
iterated_graph_orders = [1]
I used [Zyc1] equation (31) to obtain the values for $F(2^k)$ bifurcations. The code is as follows: