Tent illustration

The tent mapping is a mathematical function with the definition and value range . It is one of the simplest functions with the help of which the chaotic dynamics of nonlinear deterministic maps can be examined and, in particular, the core statement of the butterfly effect can be verified, that any small changes in the initial parameters can have large effects. ${\displaystyle \textstyle T_{2}}$ ${\displaystyle \textstyle [0,1]}$

Definition and characteristics

Graphic representation of the tent image . ${\displaystyle \textstyle T_{2}}$

The tent image is defined by:

${\displaystyle T_{2}\colon [0,1]\rightarrow [0,1],\;x\mapsto {\begin{cases}2x,&{\text{wenn }}x\in [0,{\frac {1}{2}}]\\2-2x,&{\text{wenn }}x\in {]{\frac {1}{2}},1]}\end{cases}}}$

Fixed points and periodic points

For the function maps the input value to itself. Furthermore, it follows from the structure of the function that all that can be represented as with will reach the fixed point after iterations at the latest . There is for each periodic points with the Primperiode where the fold repeated use of the initial value leads ${\displaystyle \textstyle x\in \{0,{\frac {2}{3}}\}}$ ${\displaystyle \textstyle x_{0}\in \mathbb {R} }$ ${\displaystyle \textstyle x_{0}={\frac {a}{2^{n}}}}$ ${\displaystyle a\in \{0,1,\dotsc ,2^{n}\}}$ ${\displaystyle n+1}$ ${\displaystyle \textstyle 0}$ ${\displaystyle \textstyle n\in \mathbb {N} }$ ${\displaystyle \textstyle x_{n}}$ ${\displaystyle \textstyle n}$ ${\displaystyle n}$ ${\displaystyle \textstyle T_{2}}$ ${\displaystyle \textstyle x_{n}}$

${\displaystyle x_{n}\in {\begin{cases}\{0,{\frac {2}{3}}\},&{\text{wenn }}n=1\\\{{\frac {2}{5}},{\frac {4}{5}}\},&{\text{wenn }}n=2\\\{{\frac {2}{9}},{\frac {2}{7}},{\frac {4}{9}},{\frac {4}{7}},{\frac {6}{7}},{\frac {8}{9}}\},&{\text{wenn }}n=3\\\{\dots \},&{\text{wenn }}n\in \mathbb {N} \\\end{cases}}}$

Demonstration of the butterfly effect

If you apply the tent illustration -fach one after the other to an initial value , you get a new illustration : ${\displaystyle \textstyle T_{2}(x)}$ ${\displaystyle \textstyle n}$ ${\displaystyle \textstyle x_{0}}$ ${\displaystyle \textstyle F_{x_{0}}}$

${\displaystyle F_{x_{0}}\colon \mathbb {N} \rightarrow [0,1],\;n\mapsto T_{2}^{n}(x_{0})}$

Difference in the values ​​of for and plotted against the number of iterations . After just a few iterations, the difference between the initial states can practically no longer be predicted by a perturbative consideration of the small difference in the initial value. ${\displaystyle \textstyle F_{x}(n)}$ ${\displaystyle \textstyle x_{0}=0{,}506}$ ${\displaystyle \textstyle x_{0}=0{,}506127}$ ${\displaystyle \textstyle n}$

If one compares the values ​​of for two arbitrarily close to one another , one finds arbitrarily large differences in the interval with sufficiently large differences within the value range . ${\displaystyle \textstyle F_{x}(n)}$ ${\displaystyle \textstyle x}$ ${\displaystyle \textstyle n}$ ${\displaystyle (0,1)}$

See also

Triangle function

Individual evidence

1. ↑ Julio R. Hasfura-Buenaga, Phillip Lynch: Periodic points of the Family of Tent Maps. (pdf) Retrieved March 23, 2017 (English).  

Web link

Teaching material for tents from the University of Mainz, accessed on July 17, 2018