Image:Tent-map.png

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[edit] Summary

La bildo estas kopiita de wikipedia:en. La originala priskribo estas:

Symbolic orbits of the tent map, 640x480 pixels.

[edit] Binary value of dyadic string of tent map orbits

In mathematics, and in particular, in the theory of dynamical systems, one has the concept of the symbolic orbit of a point. This graph illustrates the symbolic orbit of the unit-height tent map, expressed as a binary value.

The unit-height tent map is given by the equation:

S_\mathrm{tent}(x) = \begin{cases} 2x & \mbox{for } 0 \le x < \frac{1}{2} \\ 2-2x & \mbox{for } \frac{1}{2} \le x < 1 \end{cases}

Picking an initial value for x, and iterating on this function, one gets the orbit for the point x. This orbit can be represented symbolically as a string of letters L and R, indicating whether the nth iterate fell to the left or to the right of 1/2. By the xxx theorem, the symbolic orbits are in one-to-one correspondance with the initial value x.

Since the orbit string consists only of two symbols, these two symbols can be taken to be the binary values 0 and 1; thus, the symbolic orbit string can also be interpreted as a real number expressed in binary. This graph shows the value of this orbit string as a function of x.

More precisely, let:

sn + 1 = Stent(sn)

with s0 = x, so that the sn are the iterates of the tent map. The let

b_n = \begin{cases} 0 & \mbox{for } x < 1/2 \\ 1 & \mbox{for } x \ge 1/2 \end{cases}

Finally, let:

T(x)= \sum_{n=0}^\infty b_n / 2^n

The graph shown is a graph of T(x).

Note that this mapping is only possible for tent maps of unit height or less. Taller maps will have orbits that require more than two symbols to express.

[edit] Source

Created by Linas Vepstas User:Linas <linas@linas.org> on 25 June 2005 using custom software written entirely by Linas Vepstas.

date/time username edit summary
19:06, 20 February 2006 en:User:RussBot (Robot-assisted disambiguation (<a href="/wiki/WP:DPL" title="WP:DPL">you can help!</a>): Binary)
18:02, 25 June 2005 en:User:Linas (tweaks, remove cat)
17:58, 25 June 2005 en:User:Linas (more text,explanation, copyright status)
17:47, 25 June 2005 en:User:Linas (provide explanation)
17:39, 25 June 2005 en:User:Linas (Binary value of dyadic string of tent map orbits.)

[edit] Historio de la dosiero

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  • (del) (cur) 17:39, 25 June 2005 . . en:User:Linas Linas ( en:User_talk:Linas Talk) . . 640x480 (5292 bytes) (Binary value of dyadic string of tent map orbits.)

[edit] Licensing

GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License".

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