File:Time series of the Tent map for the parameter m=2.0 which shows numerical error.svg
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Summary
| Description |
English: Time series of the Tent map for the parameter m=2.0 which shows numerical error. This is decribed in Math Stack Exchange: "the plot of time series (plot of x variable with respect to number of iterations) stops fluctuating and no values are observed after n=50". Parameter m= 2.0, initial point is random. |
| Date | |
| Source | Own work |
| Author | Adam majewski |
| Other versions |
|
| SVG development |  This plot uses embedded text that can be easily translated using a text editor. |
Long description
The problem: "the plot of time series (plot of x variable with respect to number of iterations) stops fluctuating and no values are observed after n=50".
Parameter m= 2.0, initial point is random.
Explanation
"You are experiencing an unfortunately interplay between computer numerics and dynamical systems.
In a binary representation x the signicand part comes with a certain number of bits. Multiplying by two (and mapping the result back into [0,1]) shifts those bits, and the last bit becomes zero. When the significand at the starts is represented by 50 bits it becomes identically zero after 50 iterations.
This happens for a couple of dynamical systems where chaos comes from multiplication by 2, 4 ,8,...
The tent map as defined by f(x)=1−|2x−1|, on [0,1] also works this way by successively putting least significant digit to zero and shifting to the left.
Other examples are:
- f(x)=4xmod1 (which removes two digits at the time, so it gets constant within 25 iterations only),
- or a more complicated one:
![{\displaystyle f(x)={\begin{cases}4x\qquad \qquad x\in [0,1/4)\\2-4x\qquad x\in [1/4,1/2)\\2x-1\qquad x\in [1/2,1]\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ade21f5dc581b652289acbc25f37dce4aaf4b636)
Again at each iteration at least one more least significant binary digit becomes zero.
On the other hand, f(x)=3xmod1 is chaotic in binary representation (at least sufficiently for you to notice the computer errors).
Similarly f(x)=4x(1−x) on [0,1] is chaotic also in binary arithmetics. Indeed multiplying by 4 sets the two last digits to zero, but the product x(1−x) will make these last two digits 'random' again. So apart from the last two digits (being zero) the numbers obtained on computer from xn+1=4xn(1−xn) will appear random (with respect to a certain calculable distribution) when starting from a random initial number in (0,1).
Solutions
- See paper : Digitally generating true orbits of binary shift chaotic maps and their conjugates by Ismail Ozturk, Recai Kilic
- increase precision
Maxima Cas src code
Maxima CAS src code
/*
https://math.stackexchange.com/questions/2453939/is-this-characteristic-of-tent-map-usually-observed
*/
kill(all);
remvalue(all);
ratprint:false; /* a message informing the user of the conversion of floating point numbers to rational numbers is displayed. */
display2d:false;
/* ---------- functions ---------------------------------------------------- */
/* https://en.wikipedia.org/wiki/Tent_map */
f(x) := if x<0.5 then m*x else m*(1-x) $
GivePoints(x0, iMax):= block(
[Points,i,Point],
i:0,
x:x0,
Point:[i,x],
Points:[Point],
while (i<iMax)
do
(
x:f(x),
i:i+1,
Point:[i,x],
Points:endcons(Point, Points)
),
return(Points)
)$
/* const */
m: 2.0;
x0:random (1.0);
iMax:100;
Points:GivePoints(x0,iMax);
/* ------- draw --------------------------------------- */
load(draw);
path:"~/maxima/batch/tent/t1/"$ /* pwd, if empty then file is in a home dir , path should end with "/" */
draw2d(
user_preamble="set key top right; unset mouse; ",
terminal = 'svg,
file_name = sconcat(path,"p", string(m),"_", string(x0)),
font_size = 30,
font = "Liberation Sans", /* https://commons.wikimedia.org/wiki/Help:SVG#Font_substitution_and_fallback_fonts */
title= "Time series of the Tent map for the parameter m=2.0 ",
xlabel = "iteration ",
ylabel = "x",
dimensions = [1000, 1000],
yrange=[0,1],
color= blue,
point_type = filled_circle,
point_size = 0.2,
points_joined =true,
key = "",
points(Points)
)$C code
#include <stdio.h>
#include <stdlib.h>
#include <time.h>
/*
https://math.stackexchange.com/questions/2453939/is-this-characteristic-of-tent-map-usually-observed
gcc t.c -Wall
a@zelman:~/c/varia/tent$ ./a.out
*/
/* ------------ constans ---------------------------- */
double m = 2.0; /* parameter of tent map */
double a = 1.0; /* upper bound for randum number generator */
int iMax = 100;
/* ------------------- functions --------------------------- */
/*
tent map
https://en.wikipedia.org/wiki/Tent_map
*/
double f(double x0, double m){
double x1;
if (x0 < 0.5)
x1 = m*x0;
else x1 = m*(1.0 - x0);
return x1;
}
/* random double from 0.0 to a
https://stackoverflow.com/questions/13408990/how-to-generate-random-float-number-in-c
*/
double GiveRandom(double a){
srand((unsigned int)time(NULL));
return (((double)rand()/(double)(RAND_MAX)) * a);
}
int main(void){
int i = 0;
double x = GiveRandom(a); /* x0 = random */
for (i = 0; i<iMax; i++){
printf("i = %3d \t x = %.16f\n",i, x);
x = f(x,m); /* iteration of the tent map */
}
return 0;
}
Text code
result:
i = 0 x = 0.1720333817284710
i = 1 x = 0.3440667634569419
i = 2 x = 0.6881335269138839
i = 3 x = 0.6237329461722323
i = 4 x = 0.7525341076555354
i = 5 x = 0.4949317846889292
i = 6 x = 0.9898635693778584
i = 7 x = 0.0202728612442833
i = 8 x = 0.0405457224885666
i = 9 x = 0.0810914449771332
i = 10 x = 0.1621828899542663
i = 11 x = 0.3243657799085327
i = 12 x = 0.6487315598170653
i = 13 x = 0.7025368803658694
i = 14 x = 0.5949262392682613
i = 15 x = 0.8101475214634775
i = 16 x = 0.3797049570730451
i = 17 x = 0.7594099141460902
i = 18 x = 0.4811801717078197
i = 19 x = 0.9623603434156394
i = 20 x = 0.0752793131687213
i = 21 x = 0.1505586263374425
i = 22 x = 0.3011172526748851
i = 23 x = 0.6022345053497702
i = 24 x = 0.7955309893004596
i = 25 x = 0.4089380213990808
i = 26 x = 0.8178760427981615
i = 27 x = 0.3642479144036770
i = 28 x = 0.7284958288073540
i = 29 x = 0.5430083423852921
i = 30 x = 0.9139833152294159
i = 31 x = 0.1720333695411682
i = 32 x = 0.3440667390823364
i = 33 x = 0.6881334781646729
i = 34 x = 0.6237330436706543
i = 35 x = 0.7525339126586914
i = 36 x = 0.4949321746826172
i = 37 x = 0.9898643493652344
i = 38 x = 0.0202713012695312
i = 39 x = 0.0405426025390625
i = 40 x = 0.0810852050781250
i = 41 x = 0.1621704101562500
i = 42 x = 0.3243408203125000
i = 43 x = 0.6486816406250000
i = 44 x = 0.7026367187500000
i = 45 x = 0.5947265625000000
i = 46 x = 0.8105468750000000
i = 47 x = 0.3789062500000000
i = 48 x = 0.7578125000000000
i = 49 x = 0.4843750000000000
i = 50 x = 0.9687500000000000
i = 51 x = 0.0625000000000000
i = 52 x = 0.1250000000000000
i = 53 x = 0.2500000000000000
i = 54 x = 0.5000000000000000
i = 55 x = 1.0000000000000000
i = 56 x = 0.0000000000000000
i = 57 x = 0.0000000000000000
i = 58 x = 0.0000000000000000
i = 59 x = 0.0000000000000000
i = 60 x = 0.0000000000000000
i = 61 x = 0.0000000000000000
i = 62 x = 0.0000000000000000
i = 63 x = 0.0000000000000000
i = 64 x = 0.0000000000000000
i = 65 x = 0.0000000000000000
i = 66 x = 0.0000000000000000
i = 67 x = 0.0000000000000000
i = 68 x = 0.0000000000000000
i = 69 x = 0.0000000000000000
i = 70 x = 0.0000000000000000
i = 71 x = 0.0000000000000000
i = 72 x = 0.0000000000000000
i = 73 x = 0.0000000000000000
i = 74 x = 0.0000000000000000
i = 75 x = 0.0000000000000000
i = 76 x = 0.0000000000000000
i = 77 x = 0.0000000000000000
i = 78 x = 0.0000000000000000
i = 79 x = 0.0000000000000000
i = 80 x = 0.0000000000000000
i = 81 x = 0.0000000000000000
i = 82 x = 0.0000000000000000
i = 83 x = 0.0000000000000000
i = 84 x = 0.0000000000000000
i = 85 x = 0.0000000000000000
i = 86 x = 0.0000000000000000
i = 87 x = 0.0000000000000000
i = 88 x = 0.0000000000000000
i = 89 x = 0.0000000000000000
i = 90 x = 0.0000000000000000
i = 91 x = 0.0000000000000000
i = 92 x = 0.0000000000000000
i = 93 x = 0.0000000000000000
i = 94 x = 0.0000000000000000
i = 95 x = 0.0000000000000000
i = 96 x = 0.0000000000000000
i = 97 x = 0.0000000000000000
i = 98 x = 0.0000000000000000
i = 99 x = 0.0000000000000000
Licensing
I, the copyright holder of this work, hereby publish it under the following license:
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
File history
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 18:31, 30 June 2019 | | 1,000 × 1,000 (21 KB) | Soul windsurfer | better title |
| 18:44, 28 June 2019 |  | 1,000 × 1,000 (21 KB) | Soul windsurfer | better title | |
| 18:42, 28 June 2019 |  | 1,000 × 1,000 (21 KB) | Soul windsurfer | User created page with UploadWizard |
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