(* You should be able to copy and paste this text file directly into Mathematica.  These notes will be interpreted as commented code. *)


(* For problem 1: transcriptions of the functions *)

f1a[z_] := z + z^2 (z-I)(z+1)
f1b[z_] := z^2 - 3z + 5
f1c[z_] := z^2 - 2 I z + (I-1)
f1d[z_] := I z^2 + z + I/4
f1e[z_] := 2z^3 + 2z
f1f[z_] := 1 + 4I / (z + 2 - 3I)


(* For problem 2: transcriptions of the functions *)

f2a[z_] := 1 - I + I z
f2b[z_] := z^2 + I
f2c[z_] := 1 - (3/2) z^2 - (1/2) z^3
f2d[z_] := 3z + 4/z
f2e[z_] := z^2


(* For problem 3: transcriptions of the functions *) 

f3a[z_] := z^2 - 0.4 - 0.1 I
f3b[z_] := z^2 + 0.2 - I
f3c[z_] := z^2 + 1. I
f3d[z_] := z^2 - 0.53 + 0.6 I


(* For problem 4: transcriptions of the functions *)

f4a[z_] := z^3 + 0.4z - I
f4b[z_] := (z^3 - 1.) / (z + I)
f4c[z_] := z - (z^3 - 1.) / (3 z^2)
f4d[z_] := 2z^5 - (1+I) z^3 / 4 + (1-I)/2
f4e[z_] := (3z^3 - 2 I z + 2) / (z^2 + 3)





(* For problems 3-5: Code for invoking Mathematica's Julia set plotting algorithms, for problems 3-5 *)

f3a[z_] := z^2 - 0.4 - 0.1 I

JuliaSetPlot[f3a, z, PlotLegends -> Automatic, ImageResolution -> 200]

(* This will plot the filled Julia set for a function with default escape-time colorings, and also display the legend for comparing colorings to the number of iterates required for escape *)



JuliaSetPlot[f3a, z, ColorFunction -> None, ImageResolution -> 200]    
      
(* This will plot the Julia set for a function, with default blue points *)



JuliaSetPlot[f3a, z, ColorFunction -> (If[#3 > 9/10, RGBColor[0.5, 0.19, 0.85], White] &), ImageResolution -> 200]

(* This will plot the filled Julia set for a function, with points colored in purple *)
(* This method will not be effective for plotting Julia sets that are very sparse *)


ListPlot[JuliaSetPoints[f3a, z, "ClosenessTolerance" -> 0.005],  PlotStyle -> Directive[RGBColor[0.5, 0.19, 0.85], PointSize[0.005]], Axes -> False, Frame -> True]

(* This will plot the Julia set for a function with purple points.  To increase the resolution of the plot, decrease the value of "ClosenessTolerance" -- note that this method may take a long time if the Julia set is complicated, so it is recommended you first try plotting the set with ClosenessTolerance 0.005 and then lower the value as needed *)






(* For problem 3: Code for plotting a Julia set for a quadratic map using backwards iteration (the chaos game procedure)  *)


h1[c_][z_] := N[Sqrt[z - c]]
h2[c_][z_] := N[-Sqrt[z - c]]
hch[c_][z_] := RandomChoice[{1, 1} -> {h1[c], h2[c]}][z]

(* This defines the two components of the inverse map and a function which randomly chooses one of the maps to apply *)


julpoints[c_, n_] := Drop[NestList[hch[c], 0.01, n + 1000], 1000] 

(* This will compute n+1000 backwards iterates then delete the first 1000 terms, to clean up any slow convergence at the beginning *)


julplot[c_, n_, pointsize] := 
 ListPlot[complexlisttopointlist[julpoints[c, n]], Axes -> True, AxesLabel -> {Re, Im}, 
  PlotStyle -> Directive[RGBColor[0.64, 0.21, 0.76], PointSize[pointsize]], 
  AspectRatio -> 1, PlotLabel -> "Julia set for q(z)=z^2+(", ToString[c], ") plotted with ", ToString[n], " total points"]]

(* This will plot n points on the Julia set for q(z)=z^2+c, labeled with the parameter c and the number of points.  It may be advisable to change the point size if more points are plotted to create a higher-resolution picture. *)


julplot[-0.5, 20000, 0.0010]

(* A plot of the Julia set for z^2 - 0.5 with 20000 points and point size of 0.0010. *)






(* For problem 3: Code for looking at the critical orbit for a quadratic map *)

q[c_][z_] := z^2 + c
c3a := 0.4 - 0.1 I

(* Defining functions for ease of use *)


NestList[q[c3a], 0.0, 10]

(* This will compute the first ten terms on the critical orbit, which you can use to get an idea of the behavior *)


Drop[NestList[q[c3a], 0.0, 1030],1000]

(* This will compute the 1001st through 1030th terms on the orbit, which will generally give a very clear picture of the asymptotic behavior *)