Your personal Julia set

(10 pts)

Plot the Julia set of your personal poynomial and indicate the location of any attractive behavior that you see.

Here's the code I used to generate my Julia set:

f[z_] = -531z^3 + (48321/10)z^2-(146556/10)z+148179/10;
pic = JuliaSetPlot[f[z],z,ImageResolution->1000,
  PlotRange -> {{2.95,3.15},{-0.03,0.03}},
  Epilog -> {
    {PointSize[Medium],Green,Point[{{3,0},{3.075,0}}]},
    {PointSize[Medium],Red,Point[{3.025,0}]}}]

I found the fixed points using this code: NSolve[f[z] == z, z]

Here's the code I used to generate my Julia set:

f[z_] := (2760 + 4/10) - (8164 + 8/10) z + (8051 + 4/10) z^2 - 2646 z^3;
JuliaSetPlot[
 f[z],
 z,
 PlotRange -> {
   {0.97, 1.06},
   {-0.02, 0.02}
  },
 ImageResolution -> 1100,
 Epilog -> {
   {PointSize[Medium], Green, Point[{z0[[1]], 0}]},
   {PointSize[Medium], Red, Point[{z0[[2]], 0}]},
   {PointSize[Medium], White, Point[{z0[[3]], 0}]}
  }
]

The fixed points are $z_1=0$, $z_2=1.01241$ and $z_3=1.03044$.

$z_1$ is a super-attractive fixed point since $f'(z_1)=0$.

$z_2$ is a repulsive fixed point since $|f'(z_2)|=1.5922>1$.

$z_3$ is an attractive fixed point since $0<|f'(z_3)|=0.452204<1$.

EDIT: Here's the picture.

Here is the code I used to generate my picture

f[x_] := 23225.2 - 34591.2 x + 17173.8 x^2 - 2842 x^3

JuliaSetPlot[f[x], x, PlotRange -> {{1.98, 2.050}, {-0.015, 0.015}}, 
ImageResolution -> 1000, 
Epilog -> { 
          {PointSize[Medium], Blue, Point[{{2, 0}, {2.031788049679393`, 0}}]},
          {PointSize[Medium], Red, Point[{2.011069093162045`, 0}]}}]

and here is my picture

JuliaSet (2).png739x571 45.9 KB

I used this code Solve[f[x] == x, x] to get the fixed points x=1, 1.10918, and 1.19082.
I used the following code to generate my Julia Set:
f[x_] := 63.4 \[Minus] 172.8 x + 158.4 x^2 \[Minus] 48 x^3;
JuliaSetPlot[f[x], x, PlotRange -> {{0.8, 1.4}, {-0.16, 0.16}},
ImageResolution -> 1100,
Epilog -> {{PointSize[Medium], Blue,
Point[{{1, 0}, {1.19082, 0}}]}, {PointSize[Medium], Red,
Point[{1.10918




, 0}]}}]`

Here is the picture generated by Mathematica:

My Personal (complex) Polynomial is $f(z)=14315.7-14158.8z+4668.3z^2-513z^3$.

I used the code

NSolve[f[z] == z, z]

To find the fixed points $3$,$3.02653$, and $3.07347$. To classify these points, I calculated
$|f'(3)|=0 \Rightarrow 3$ is a super attractive fixed point,
$|f'(3.02653)|=1.63877>1$, so $3.02653$ is a repulsive fixed point,
$|f'(3.07347)|=.769255$, so $3.07347$ is an attractive fixed point.

Having found these characteristics of our set, we now use the following code to generate the following image:

pic = JuliaSetPlot[f[z], z, ImageResolution -> 1000,
PlotRange -> {{2.95, 3.11}, {-0.03, 0.03}},
   Epilog -> {
	{PointSize[Medium], Green, Point[{3.0000000000036087, 0}]},
    {PointSize[Medium], Red, Point[{3.02653, 0}]},
{PointSize[Medium], White, Point[{3.07347, 0}]}
}
]

My polynomial is $f(z) = 115.5 z^2 - 2695 z^3$, I plotted its Julia set with the following Mathematica Code

f[z_] = 115.5 z^2 - 2695 z^3;
pic = JuliaSetPlot[f[z], z, ImageResolution ->     1000, 
  PlotRange -> {{-.02, .05}, {-0.02, 0.02}}, 
  Epilog -> {{PointSize[Medium], Green, 
  Point[{{0, 0}, {0.012041, 0}}]},     {PointSize[Medium], Red, 
 Point[{0.0308161, 0}]}}]

The plot can be seen below with a super attractive fixed point at $0$ an attractive fixed point at $x =.0308161$
and a repulsive fixed point at $x =0.012041$

I used this code to generate my Julia set:

f[z_] = 19621.6 - 29223.6 z + 14508.9 z^2 - 2401 z^3;
JuliaSetPlot[f[z], z, PlotRange -> {{1.98, 2.05}, {-.015, .015}},<img 
ImageResolution -> 1100, 
Epilog -> {{PointSize[Medium], Green, 
Point[{{2.01489, 0}, {2.02796, 0}}]}, {PointSize[Medium], Red, 
Point[{2.00, 0}]}}]

This pretty thang has a super attractive fixed point at
z=2, a repulsive fixed point at z = 2.01489, and an attractive fixed point at
z = 2.02796.

I used this code to generate my Julia Set:

f[z_] = -406 - 1260 z - 1305 z^2 - 450 z^3
pic = JuliaSetPlot[f[z], z, ImageResolution -> 1000, 
PlotRange -> {{-1.05, -.89}, {-0.03, 0.03}}, 
Epilog -> {{PointSize[Medium], Green, 
 Point[{-1, 0}]}, {PointSize[Medium], Red, 
 Point[{-0.9666666666667537, 0}]}, {PointSize[Medium], White, 
 Point[{-0.9333333333332109, 0}]}}]

The point (-1,0) is a superattractive fixed point, as f'[-1] = 0.
The point (-0.96666,0) is a repulsive fixed point, as f'[-0.96666] = 1.5, which is greater than 1.
the point (-0.93333,0) is a superattractive fixed point, as f'[-0.9333333333332109] = 0.

I checked my 2 fixed points for behavior, one was super attractive, one attractive, and one repulsive and one.The super attractive point is at $z=-1$. The attractive point $z=-0.934754$ has multiplier $\lambda=0.122672$, and is quite close to the critical point at $z=-.933\bar{3}$. The repulsive point at $z=-0.965246$ has multiplier $\lambda=1.46733$. I use the following code:

f[x_] = -3979/10 - 12348/10 x - 12789/10 x^2 - 441 x^3;
NSolve[f'[x] == 0, x]
NSolve[f[x] == x, x]
f'[-0.9652455338982358`]
f'[-0.9347544661016687`]
f[-0.9333333333333309`]

Then I used the following code to generate a julia set plot with a green dot at the super attractive point, a blue dot at the attractive point, and a red dot at the repulsive one:

JuliaSetPlot[f[x], x, ImageSize -> Large, 
 PlotRange -> {{-1.045, -.89}, {-.05, .05}}, 
 Epilog -> {{PointSize[Medium], Green, 
    Point[{{-1, 0}}]}, {PointSize[Medium], Blue, 
    Point[{{-0.9347544661016687`, 0}}]}, {PointSize[Medium], Red, 
    Point[{-0.9652455338982358`, 0}]}}]

For some fun stuff:

Using

g[z_] = -41*z^3 + 381.3*z^2 - 1180.8*z + 1220.7;
JuliaSetPlot[g[z], z,
ImageSize -> Large,
Epilog -> {{PointSize[Medium], Green, 
Point[{{3, 0}}]}, {PointSize[Medium], Blue, Point[{{3.2, 0}}]}}]

I generated the picture

There are two attractive fixed points, one is located at 3.2 and the other is located at 3 and is super-attractive!

@notneds That's crazy looking!