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# XerxesZorgon/JuliaSetPlots.jl

Created February 25, 2022 01:40
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Plots trajectories and filled Julia sets
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 #= Plot Julia functions Load with: include("JuliaSetPlots.jl") Written by: John Peach 13-Feb-2022 Wild Peaches Julia sets are defined as z_{n+1} = zₙ² + c where zₙ ∈ ℂ and c = x₀ + iy₀ Mandelbrot sets are similar, but z₀ is fixed, and c ∈ ℂ. =# # Required packages using Plots, LinearAlgebra gr() """ plotIterates Plots iteratations of the Julia function from in initial point z₀ ∈ ℂ and parameter c Parameters z₀: Starting point c: Control parameter n: Number of iteratations Returns Plot of n iteratations """ function plotIterates(z₀,c,n) # Initialize trajectory zₙ = length(z₀) traj = zeros(ComplexF64,n,zₙ) traj[1,:] .= z₀ # Generate trajectory for k = 1:zₙ for j = 2:n traj[j,k] = traj[j-1,k]^2 + c end end # Plot trajectory. The display function is needed inside a function. crvLabel = Array{String}(undef,zₙ) for k = 1:zₙ crvLabel[k] = string("z₀ = ", z₀[k]) end # Note permation of label string. # Ref: https://discourse.julialang.org/t/transpose-of-an-array-of-strings/40431/8 crvPlot = plot(traj, linewidth = 3, aspect_ratio = 1, xlabel = "Real axis", ylabel = "Imag axis", title = "Julia Iterates", arrow = :arrow) #label = permutedims(crvLabel) #legend = :outertopright) display(crvPlot) # Return trajectory return traj end """ plotJulia Plots points in a Julia set Parameters c: Constant added to each iterate """ function plotJulia(c,xRng = [-1.6,1.6], yRng = [-1.6, 1.6]) # Array of iteration counts nx = 1281 ny = 1281 iterCounts = zeros(nx,ny) # Locations xLocs = range(minimum(xRng),stop = maximum(xRng),length = nx) yLocs = range(minimum(yRng),stop = maximum(yRng),length = ny) # Loop over x,y locations iterating until maxIter reached, or |z|² > 4 maxIters = 1000 for j = 1:nx for k = 1:ny z = xLocs[j] + yLocs[k]im while iterCounts[j,k] < maxIters && norm(z) < 2 z = z^2 + c iterCounts[j,k] += 1 end end end # Display Julia set as a heatmap display(heatmap(iterCounts')) # Return array of iterations return iterCounts end