File:Bifurcation diagram for real quadratic map. Periodic points for periods 1,2,4,and 8 are shown.png

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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

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Software used[edit]

 
This plot was created with Gnuplot.
  • Maxima CAS
    • package Draw by Mario Rodriguez Riotorto[1]
  • Image Magic
  • gedit

Summary[edit]

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[edit]

/*

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[edit]

convert -resize 1000x1000 23.svg 23.png

References[edit]

  1. ↑ Mario Rodriguez Riotorto - A Maxima-Gnuplot interface archive copy at the Wayback Machine
  2. ↑ Bifurcation and Orbit Diagrams by Chip Ross

File history

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current15:24, 26 February 2014Thumbnail for version as of 15:24, 26 February 20141,000 × 1,000 (404 KB)Soul windsurfer (talk | contribs)title
15:09, 26 February 2014Thumbnail for version as of 15:09, 26 February 20141,000 × 1,000 (404 KB)Soul windsurfer (talk | contribs)User created page with UploadWizard

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