First Favorite Polynomial
My first favorite cubic polynomial is f(z)=z^3-z. f_{1}(z) has a connected Julia set, so for g_{1}(z), we let M=1.
-
g_{1}(z)=z^3-z
-
The critical points of g_{1} are where g'_{1}(z)=0.
g'_{1}(z)=3z^2-1
3z^2-1=0
z^2=\frac{1}{3}
So there are critical points at \sqrt{\frac{1}{3}} and -\sqrt{\frac{1}{3}}.
I verified this with Mathematica using the following code:
g1 = z^3 - z;
dg1 = D[g1, z];
Solve[dg1 == 0, z]
NSolve[dg1 == 0, z]
Which gives us:
{{z -> -(1/Sqrt[3])}, {z -> 1/Sqrt[3]}}
{{z -> -0.57735}, {z -> 0.57735}}
This is what we got when solving it by hand.
g_{1} has two critical points at \sqrt{\frac{1}{3}} and -\sqrt{\frac{1}{3}}.
- Fixed points of g_{1} are the points that satisfy g_{1}(z)=z.
g_{1}(z)=z^3-z=z
z^3-2z=0
z(z^2-2)=0
This is satisfied for z=0, z=\sqrt{2}, and z=-\sqrt{2}.
I verified this with Mathematica as well:
g1 = z^3 - z;
Solve[g1 == z, z]
NSolve[g1 == z, z]
Which gave us:
{{z -> 0}, {z -> -Sqrt[2]}, {z -> Sqrt[2]}}
{{z -> -1.41421}, {z -> 0.}, {z -> 1.41421}}.
This verifies our previous answer.
0, \sqrt{2}, and -\sqrt{2} are the fixed points of g_{1}.
-
The plot of the Julia set of g_{1} and the orbits generated by g_{1} starting from \sqrt{\frac{1}{3}} and -\sqrt{\frac{1}{3}}:
-
The orbits seem to converge onto the fixed point at the origin, and the orbits bounce back and forth across the imaginary axis while remaining on the real axis. Neither orbit converges onto the fixed points at \sqrt{2} or -\sqrt{2}.
To be sure of this, I decided to classify the fixed points by finding the value of |g'_{1}(z)| at each of the fixed points:
|g'_{1}(\sqrt{2})|=|3(\sqrt{2})^2-1|=|3(2)-1|=|5|>1, so \sqrt{2} is a repulsive fixed point.
|g'_{1}(-\sqrt{2})|=|3(-\sqrt{2})^2-1|=|3(2)-1|=|5|>1, so -\sqrt{2} is a repulsive fixed point.
|g'_{1}(0)|=|3(0)^2-1|=|-1|=1, the origin is a neutral fixed point.
Because the fixed point at 0 is not attractive, I decided to iterate the function starting at each of the critical points a few times just to get a better picture of how it may be converging onto the origin. I did this iteration in Mathematica:
g1[z_] = z^3 - z;
Table[N[Nest[g1, Sqrt[1/3], n]], {n, 1, 15}]
Table[N[Nest[g1, - Sqrt[1/3], n]], {n, 1, 15}]
This gave these lengthy outputs:
{-0.3849, 0.327878, -0.29263, 0.267571, -0.248415, 0.233085, -0.220422, 0.209712, -0.200489, 0.192431, -0.185305, 0.178942, -0.173212, 0.168015, -0.163272} for \sqrt{\frac{1}{3}}.
{0.3849, -0.327878, 0.29263, -0.267571, 0.248415, -0.233085, 0.220422, -0.209712, 0.200489, -0.192431, 0.185305, -0.178942, 0.173212, -0.168015, 0.163272} for -\sqrt{\frac{1}{3}}.
They have not gotten very close to the origin yet, but my computer really didn’t want to do any more than 15 iterations. I think that it is pretty safe to assume that with more iteration, the orbits will continue to converge on the fixed point 0.
The orbits converge on the fixed point at 0, but not on the fixed points at \sqrt{2} and -\sqrt{2}.
Second, and also Favorite, Polynomial
All of this was previously written as a reply to my original post, but I figured I shouldn’t take up that extra space on this forum so I’ll just edit it in here:
At the beginning of the semester, I actually put two polynomials, so I figured its only fair if I give them similar treatments (I also don’t have anything to do today, so I figured I might as well do this to pass sometime). I’ll keep this much shorter than my first one. The second polynomial was f_{2}(z)=z^3+z and it already has a connected Julia set as well. Since M=1,
g_{2}(z)=z^3+z.
For the critical points, I used mathematica:
g2 = z^3 + z;
dg2 = D[g2, z];
Solve[dg2 == 0, z]
NSolve[dg2 == 0, z]
which gave us
{{z -> -(I/Sqrt[3])}, {z -> I/Sqrt[3]}}
{{z -> 0. - 0.57735 I}, {z -> 0. + 0.57735 I}}
g_{2} has critical points at i\sqrt{\frac{1}{3}} and -i\sqrt{\frac{1}{3}}.
I then also used Mathematica to find the fixed points:
g2 = z^3 + z;
Solve[g2 == z, z]
Which gave
{{z -> 0}, {z -> 0}, {z -> 0}}
The only fixed point is at the origin.
Taking a look at the plot of the Julia set or g_{2} with the orbits generated by starting at 0.57735i and -0.57735i, we can see where the orbits are converging:
It looks like this time the orbits are converging to the fixed point at 0, going straight in on the imaginary axis.
It is interesting that on g_{1}, the iterations starting on the critical points alternated between positive and negative real values; however on g_{2}, the iterations stay in the upper-half or lower-half plane that is respective to the starting critical point the whole time.
Stay safe and enjoy the snow everybody!
