Matlab chaotic, fractal

for function f (x) =λsin (πx), λ∈ (0,1], use MATLAB to calculate the value of iteration x=f (x) as λ grows, the code is as follows:

function Y=diedai (f,a,x1)

n=32;

Y=zeros (n,1);

For I=1:1e4

X2=f (A,X1);

x1=x2;

Y (mod (i,n) +1) =x2;

End

End

%[email protected] (a,x) a*x* (1-x);

[Email protected] (a,x) A*sin (pi*x);

%x0=0.1;

Hold on;

For x0=-1:0.05:1

For A=0:0.01:1

Y=diedai (f,a,x0);

For count=1:32

Plot (A,y (count), ' K. ');

Hold on;

End

End

End

The resulting image is as follows: where the horizontal axis is λ, the longitudinal axes are X

It can be seen that as λ gradually increases, there are times when the bifurcation of the cycle.

It can be seen from the figure that the first fork value is around 0.3, the second is between 0.73 and 0.75, the third is between 0.8 and 0.85, and chaos occurs around 0.86. Next write the code to calculate the fork value, the code is as follows:

Format long;

x0=0.1;

For a=0.3182:0.0000001:0.3183

Y=diedai (f,a,x0);

If Max (y) >0.001

Disp (a);

Break

End

End

Get the first fork value of about 0.3182298

Format long;

x0=0.1;

For a=0.7199:0.000001:0.72

Y=diedai (f,a,x0);

If Max (y)-min (y) >0.001

Disp (a);

Break

End

End

Get a second fork value of about 0.719911

Format long;

x0=0.1;

For a=0.8332:0.000001:0.8333

Y=diedai (f,a,x0);

If ABS (Y (+)-y ()) >0.001

Disp (a);

Break

End

End

Get a third fork value of about 0.833267

Using the Feigenbaum constant to estimate the third fork value, we get 0.805939

Fractal chart

Zhou Evergreen

Draw Mandelbrot Fractal chart, mainly uses three functions: Iter=mandelbrot1 (X0,y0,maxiter), used to calculate whether the convergence after the iteration, the equation z=z2+z0. C=color (Iter,maxiter) calculates the color value and returns [R G B]. DRAW_MANDELBROT1 is used to draw images.

function Iter=mandelbrot1 (x0,y0,maxiter)

x=x0;

Y=y0;

For I=1:1:maxiter

if (x*x+y*y) >=4

Iter=i;

Break

Else

tem=x*x-y*y+x0;

Y=x*y*2+y0;

X=tem;

End

Iter=i;

End

End

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function C=color (iter,maxiter)

If Iter==maxiter

c=[1,0,0];

Else

C1=abs (mod ((iter*20+255), 510)-255);

C2=abs (mod ((iter*15+85+255), 510)-255);

C3=abs (mod ((iter*30+171+255), 510)-255);

C=[C1/255,C2/255,C3/255];

End

End

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function draw_mandelbrot1

For Y=0:ymax

For X=0:xmax

Xt=2*r*x/xmax+x0-r;

Yr=r*ymax/xmax;

Yt=2*yr*y/ymax+y0-yr;

ITER=MANDELBROT1 (Xt,yt,maxiter);

%c=color (Iter,maxiter);

C=color (Iter,maxiter);

Plot (Xt,yt, '. ', ' color ', c);

Hold on;

End

End

End

Enter DRAW_MANDELBROT1 ( -0.5,300,0,200,2,300) and the resulting image is as follows:

The color of the convergent part is also drawn, drawing the Julia Fractal chart, mainly using three functions [Xlist,ylist,iter]=julia1 (X0,Y0,MAXITER,JX0,JY0), C=color2 (Iter,maxiter,xlist, Ylist), Draw_julia1 (X0,xmax, Y0,ymax, r,maxiter,jx0,jy0). Where julia1 uses the equation Z ' =z2+zm to iterate, z0=x0+j*y0,zm=jx0+j*jy0; returns to ITER for the number of iterations, COLOR2 based on different ITER, and X, y values, calculated c=[r G b],draw_ Julia1 the drawing.

Enter Draw_julia1 (0,300,0,200,1.6,50,-0.78888,0.212325) in MATLAB to get

The code is as follows:

function [Xlist,ylist,iter]=julia1 (X0,Y0,MAXITER,JX0,JY0)

Xlist (1) =x0;

Ylist (1) =y0;

m=256;

Lnln_m=log (ABS (log (M)));

xbck=0;

ybck=0;

x=x0;

Y=y0;

For I=1:maxiter

if (x*x+y*y>=m)

% Iter=i;

Break

End

Xbck=x;

Ybck=y;

tem=x*x-y*y+jx0;

Y=x*y*2+jy0;

X=tem;

Xlist (i+1) =x;

Ylist (i+1) =y;

End

If I~=maxiter

Lnln_z=log (ABS (log (x*x+y*y)));

Lnln_zbak=log (ABS (log (XBCK*XBCK+YBCK*YBCK)));

iter=i-2-(Lnln_z-lnln_m)/(Lnln_z-lnln_zbak);

Else

Iter=i;

End

End

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function C=color2 (iter,maxiter,xlist,ylist)

% Xlist=xylist (1:length (xylist)/2);

% ylist=xylist (Length (xylist)/2:length (xylist));

[Email protected] (x) (Sin (x*2*pi/510-pi*0.5) +1) *0.5*255;

If Iter==maxiter

X=xlist (Maxiter);

Y=ylist (Maxiter);

Z=SQRT (X*x+y*y);

Zd=z-sqrt (Xlist (maxIter-1) *xlist (maxIter-1) +ylist (maxIter-1) *ylist (maxIter-1));

R1=sincolorf (z*2000);

G1=sincolorf (y*x*1000);

B1=sincolorf (zd*1000);

% Errcolor (1) =errcolor (1) +r1;

% Errcolor (2) =errcolor (2) +g1;

% Errcolor (3) =errcolor (3) +b1;

Else

R1=sincolorf (ITER*20);

G1=sincolorf (iter*15+85);

B1=sincolorf (iter*30+171);

% Errcolor (1) =errcolor (1) +r1;

% Errcolor (2) =errcolor (2) +g1;

% Errcolor (3) =errcolor (3) +b1;

End

% Result_r=ftcolor (Errcolor (1));

% Result_b=ftcolor (Errcolor (2));

% Result_g=ftcolor (Errcolor (3));

r=r1/255;

g=g1/255;

b=b1/255;

C=[r G B];

End

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function Draw_julia1 (X0,xmax, Y0,ymax, r,maxiter,jx0,jy0)

For Y=1:ymax

For X=1:xmax

Xt=2*r*x/xmax+x0-r;

Yr=r*ymax/xmax;

Yt=2*yr*y/ymax+y0-yr;

[Xlist,ylist,iter]=julia1 (XT,YT,MAXITER,JX0,JY0);

% xylist=[xlist Ylist];

C=color2 (iter,maxiter,xlist,ylist);

Plot (Xt,yt, '. ', ' color ', c);

Hold on;

%drawnow;

End

End

End

It can be seen that by color2 the color in the convergent region can be calculated, the more colorful fractal map, but overall, because of the lack of knowledge, it is difficult to find a suitable color scheme.

Matlab chaotic, fractal