ملف:Time series of the Tent map for the parameter m=2.0 which shows numerical error.svg
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ملخص
| الوصف |
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. |
| التاريخ | |
| المصدر | عمل شخصي |
| المؤلف | Adam majewski |
| إصدارات أخرى |
|
| SVG منشأ الملف |  الشيفرة المصدرية لهذا الرسم المتجه صالحة.  هذا الرسم المتجهي أُنشئ بواسطة Gnuplot  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
ترخيص
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تاريخ الملف
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| زمن/تاريخ | صورة مصغرة | الأبعاد | مستخدم | تعليق | |
|---|---|---|---|---|---|
| حالي | 18:31، 30 يونيو 2019 | | 1٬000 × 1٬000 (21 كيلوبايت) | Soul windsurfer | better title |
| 18:44، 28 يونيو 2019 |  | 1٬000 × 1٬000 (21 كيلوبايت) | Soul windsurfer | better title | |
| 18:42، 28 يونيو 2019 |  | 1٬000 × 1٬000 (21 كيلوبايت) | Soul windsurfer | User created page with UploadWizard |
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