Mandelbrot and Julia Set Plots

Use the slider-bars to locate interesting points on the Mandelbrot set.

Manipulate[Show[{MandelbrotSetPlot[], Graphics[{PointSize[Large], Pink, Point[{a, b}]}]}, PlotLabel->{a, b}], {a, -2,2}, {b, -1.25, 1.5}]

Below is the Julia Set for the parameter c = a0+b0*i.  The long-term behavior of the orbit of f_c(0) is plotted in the final pane.  Notice that if c is in the main region, the point zero is in the basin of an attracting fixed point.  This set of parameters is called the 1-bulb.  To the left, zero is in the basin of an attracting two-periodic orbit.  This set of initial parameters called a 2-bulb.   Search for higher order bulbs in the Mandelbrot set.   Try c = -0.36 + 0.63*I.  Try c = -0.62 + 0.44*I.

a0 = 0;
b0=.03;
numberiterates = 10000;
plotfinal = 40;

f[c_][z_]:= z^2 + c
Points[a_, b_] := If[MandelbrotSetMemberQ[a + b*I], Take[NestList[f[a+b*I], 0, numberiterates], -plotfinal], {3}]

GraphicsGrid[{{Show[{MandelbrotSetPlot[], Graphics[{PointSize[Large], Pink, Point[{a0, b0}]}]}]}, {JuliaSetPlot[a0+b0*I]}, {ListPlot[{Re[#],Im[#]}&/@Points[a0, b0],AxesOrigin->{0,0},PlotRange->{{-2,2},{-1,1}},ImagePadding->40,AspectRatio->1,Frame->True,FrameLabel->{{Im,None},{Re,"complex plane"}},PlotStyle->Directive[Red,PointSize[.02]]]}}]