EXERCISE: Plot the Mandelbrot set.mandelbrot:=proc(cx,cy)
local f,i, x0;
f:=z->z^2+cx + cy*I;
x0:=0;
for i from 1 to 250 while(evalf(abs(x0))<2) do x0:=f(x0);
end do:
return i/500;
end proc:with(plots):mand:=densityplot(mandelbrot, -1.7..1.7,-1.7..1.7, style = patchnogrid, colorstyle=HUE, grid = [150, 150]):pt:=complexplot([-0.68+0.13*I,-0.186+0.743*I,-0.060+0.704*I,0.286+.54*I,-0.12+.75*I],style=point, thickness=2,color=black):
display(mand, pt);EXERCISE: For each c in the following list :
1. c:=-1
2. c:=-0.8
3. c:=-0.68+0.13*I;
4. c:=i5. c:=-0.186+0.743*I;
6. c:=-.515+.56*I;7. c:=0.12+.75*I;
8. c:=-0.060+0.704*I;
9. c=-.12-0.75*I:
10. c:==-1.1;11. c:=-1.3
12. c:=0.286+.54*I;
13 c:=-0.486+.54*I;
14 c:= -0.12+0.74*I2. Describe the fate of the orbit of 0 under f(z)= z^2+c.
3. Plot c and the mandelbrot set together.
4. Compute the corresponding filled Julia set.5. Can you describe any correlation between the fate of the orbit of 0 for each c, and the shape of hte corresponding Julia set?
6. Can you describe any correlation between the fate of the orbit of 0 for each c, and the location of c in the Mandelbrot set.mandelbrot(-1,0); # probably in the M-set
mandelbrot(-0.8,0); #probably in the M-set
mandelbrot(-0.68,0.13); #probably in the M-set
mandelbrot(0,1); #probably in the M-set
mandelbrot(-0.186,0.743); #probalby in the M-set
mandelbrot(-.515,.56); # probably in the M-set
mandelbrot(-0.12,-0.75); #probably in the M-set
mandelbrot(-1.1,0); #probably in the M-set
mandelbrot(-1.3,0); #probalby in the M-set
mandelbrot(0.286,.54); # probably in the M-set
mandelbrot(-0.486,+0.54); #probably in the M-set
mandelbrot(-0.12,0.74); #probably in the M-set2We consider c=-1. The orbit of 0 is -1, 0,-1,0,-1... 0 is a periodic point of period 2 for the function f(z)=z^2-1c:=-0.68+0.13*I; We compute the orbit of 0 under f(z)=z^2-0.68+0.13*I with a for loop.c:=-0.68+0.13*I;
f:=z->z^2+c;
x0:=0;
for i from 1 to 150 do x0:= evalf(f(x0)); end do;The orbit of 0 gets closer and closer to a fixed point.c:=-0.186+0.743*I; The orbit of 0 in this case converges to an orbit of period 3.c:=- 0.12+.75*I; The orbit of 0 in this case converges to an orbit of period 3.c:=-0.12+0.75*I;
f:=z->z^2+c;
x0:=0;
for i from 1 to 150 do x0:= evalf(f(x0)); end do;c:=-1.1;c:=-1.1;
f:=z->z^2+c;
x0:=0;
for i from 1 to 150 do x0:= evalf(f(x0)); end do;The orbit of 0 in this case converges to an orbit of period 2.EXERCISE: Plot the Mandelbrot set associated to other functions. e.g. f(z)=z^3+c; or f(z)=z^2-c.z