File:Bifurcation diagram for real quadratic map. Periodic points for periods 1,2,4,and 8 are shown.png
Size of this preview: 600 × 600 pixels. Other resolutions: 240 × 240 pixels | 480 × 480 pixels | 1,000 × 1,000 pixels.
Original file (1,000 × 1,000 pixels, file size: 404 KB, MIME type: image/png)
 |
This is a file from the Wikimedia Commons |
Summary
| Description |
English: Bifurcation diagram for real quadratic map. Periodic points for periods 1,2,4,and 8 are shown |
| Date | |
| Source | Own work |
| Author | Adam majewski |
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 3.0 Unported 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.
Software used
This plot was created with Gnuplot.
- Maxima CAS
- package Draw by Mario Rodriguez Riotorto[1]
- Image Magic
- gedit
Summary
For given :
- parameter c ( c is constant )
- period n
- map ( function) f
periodic points z are roots of the equation :[2]
so set of periodic points z is :
On the image only first 4 periods of period doubling cascade are shown : 1,2,4 and 8 ( see list DrawList )
One can improve it by finding irreducible polynomial (giving only period n points, see G function ) instead of reducible polynomial ( giving period n and its divisors, see Fn function)
Set of periodic points contains both attracting and repelling points. This is a difference with orbit diagram which shows periodic and chaotic points
Maxima CAS src code
/*
Maxima CAS batch file
finds periodic points :
* for periods 1,2,4,8
* of real quadaratic map
*/
kill(all);
remvalue(all);
/* ---------- functions ---------------------- */
/* basic funtion = monic and centered complex quadratic polynomial
http://en.wikipedia.org/wiki/Complex_quadratic_polynomial
*/
f(z,c):=z*z+c $
/* iterated function */
fn(n, z, c) :=
if n=1 then f(z,c)
else f(fn(n-1, z, c),c) $
/* roots of Fn are periodic point of fn function */
Fn(n,z,c):=fn(n, z, c)-z $
/* gives irreducible divisors of polynomial Fn[p,z,c]
which roots are periodic points of period p */
G[p,z,c]:=
block(
[f:divisors(p),t:1],
g,
f:delete(p,f),
if p=1
then return(Fn(p,z,c)),
for i in f do t:t*G[i,z,c],
g: Fn(p,z,c)/t,
return(ratsimp(g))
)$
/* use :
roots:GiveRoots_bf(G[3,z,c]);
*/
GiveSolutio(g):=
block(
[cc],
cc:realroots(expand(g)=0),
cc:map('rhs,cc),/* remove string "c=" */
cc:map('float,cc),
return(cc)
)$
GiveDrawList(Period,cMin, cMax, cStep):=
block
(
[MyList:[],P,s],
for c:cMin while c <= cMax step cStep do
(
P:G[Period,z,c],
s:GiveSolutio(P),
for z in s do MyList:cons([c,z],MyList)
),
return(MyList)
)$
compile(all);
/* ---------- constant ---------------------------*/
nMax:3$
cMin:-2;
cMax:0.25;
cStep:0.003;
/* ------------- main -----------------------*/
ColorList:[red,blue,green, black, purple , brown, cyan, violet,gray ]$
DrawLists:[]$ /* empty list */
for n:0 thru nMax do
(
print(n),
Period : 2^n,
DrawList:GiveDrawList( Period,cMin, cMax, cStep),
DrawLists:cons(points(DrawList),DrawLists),
DrawLists:cons(key =concat("period =", string(Period)),DrawLists),
DrawLists:cons(color=ColorList[n+1],DrawLists)
);
/* ----------------- draw ------------ */
path:"~/maxima/batch/1ddiagram/periodic/"$ /* result of pwd; to save image in the same directory as mac file, change it manually */
FileName:concat(string(nMax)," ", string(cStep))$ /* without extension which is the terminal name */
load(draw); /* http://riotorto.users.sourceforge.net/gnuplot/ */
draw2d(
terminal = svg,
file_name = concat(path,FileName),
title = "Periodic curves diagram for real quadratic map fc(z) = z^2 + c",
dimensions=[2000,2000],
user_preamble = "",
xlabel = "c",
ylabel = "z",
xaxis = true,
points_joined =false,
point_size = 0.2,
point_type = filled_circle,
DrawLists
);Bash and Image magic code
convert -resize 1000x1000 23.svg 23.png
References
- ↑ Mario Rodriguez Riotorto - A Maxima-Gnuplot interface archive copy at the Wayback Machine
- ↑ Bifurcation and Orbit Diagrams by Chip Ross
File history
Click on a date/time to view the file as it appeared at that time.
| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 15:24, 26 February 2014 | | 1,000 × 1,000 (404 KB) | Soul windsurfer | title |
| 15:09, 26 February 2014 |  | 1,000 × 1,000 (404 KB) | Soul windsurfer | User created page with UploadWizard |
File usage
The following 2 pages use this file:
