We now seek a value of $\alpha$ such that $f_\alpha(z)$ has four super attractive orbits.

Utilizing mathematica, we write

```
f[a_][z_] = z^2/(1 + a z^5)
F[a_][z_] = Nest[f[a], z, 4]
eqs = {F[a][z] == z, F[a]'[z] == 0}
as = a /. NSolve[eqs, {a, z}]
as = Union[as, SameTest -> (Abs[#1 - #2] < 0.0000001 &)]
as
```

This gives us a list of candidate $\alpha$ values as a list, namely ${-0.209273, -0.130092 + 0.0227936 I, -0.130092 -

0.0227936 I, -0.115265 + 0.108163 I, -0.115265 -

0.108163 I, -0.110725 - 0.0885781 I, -0.110725 +

0.0885781 I, -0.0262079 - 0.0777137 I, -0.0262079 +

0.0777137 I, -0.00939762 + 0.150968 I, -0.00939762 -

0.150968 I, -0.0090919 + 0.00613058 I, -0.0090919 -

0.00613058 I, -0.00525272 - 0.00793709 I, -0.00525272 +

0.00793709 I, -0.0038916 + 9.54729*10^-9 I, -0.00389148 -

1.18249*10^-7 I, -0.00389147 - 2.59888*10^-10 I, -0.00389145 +

1.49564*10^-7 I, -0.00389132 -

1.06469*10^-8 I, -0.00321798, 0.000302474,

0.000838845 - 0.0424907 I, 0.000838845 + 0.0424907 I, 0.0049318,

0.00978064 + 2.3801*10^-7 I, 0.00978155 + 1.59318*10^-9 I,

0.00978165 + 6.84955*10^-7 I, 0.0097823 - 1.77586*10^-7 I,

0.0479572 - 0.0569088 I, 0.0479572 + 0.0569088 I,

0.0486073 - 0.00326855 I, 0.0518073 - 5.39964*10^-6 I,

0.0518099 - 1.86444*10^-6 I, 0.0518134 - 6.74436*10^-7 I,

0.051815 + 5.80434*10^-7 I, 0.0518151 - 4.75716*10^-6 I,

0.0518323 - 0.0000344687 I, 0.0518362 - 0.0000323141 I,

0.05184 - 0.0000279354 I, 0.051841 + 0.0000264454 I,

0.051842 + 0.0000278376 I, 0.0518455 - 0.0000244157 I,

0.0518472 + 0.0000290001 I, 0.0518507 - 0.0000329779 I,

0.0518533 - 0.0000368313 I, 0.0518684 - 7.00661*10^-6 I,

0.0518691 - 0.000011351 I, 0.0518725 - 5.441*10^-6 I,

0.0548728 + 0.00301302 I, 0.0551607 - 0.0031988 I}$.

Using our rational julia set tool, we set out to test some of these $\alpha$ candidates for the desired behavior.

Let us consider $\alpha= -0.110725 - 0.0885781 i$. Plugging this value into our rational julia set tool, we generate the following graphic, which shows some interesting periodic behavior:

We again utilize mathematica to find values of $z$ which exhibit the periodic behavior.

```
F4[z_] = FP[FP[FP[FP[z]]]]
Solve[F4[z] == z, z]
JuliaSetPlot[FP[z], z]
```

This generates a relatively large list of values of $z$, and the following image, with which we use the get coordinates function to get approximations of the values of $z$ we are looking for. Armed with this, we select particular candidates from our list of $z$ values.

We settle upon the following four points as representatives for each of the four period four orbits:

Red,$(0.5302005551614561 + 0.5072329008030877 i)$,

Green: $(-0.727085 - 0.098716 i)$,

Pink $(0.646248 - 0.347507 i)$,

Yellow $(-0.130797 - 0.722004 i)$.

For each of these points, it can be seen that they return to their initial value after 4 iterations of our function. Additionally, $f'_\alpha(z)=0$ at all four of these values, and so they are super attractive orbits of period 4.

It is worth noting that this particular value of $\alpha$ also has a period $2$ orbit, denoted in blue.