JuliaSetPlot

plots the Julia set of the rational function f of the variable z.

plots the Julia set of the function .

Details and Options

The Julia set of a function f is the closure of the set of all repelling fixed points of f.
JuliaSetPlot has the same options as Graphics, with the following additions and changes:

AspectRatio	Automatic	ratio of height to width of the plot
Axes	False	whether to include axes
ColorFunction	Automatic	how to color the exterior
ColorFunctionScaling	True	whether to scale arguments to ColorFunction
Frame	True	whether to include a frame
ImageResolution	Automatic	resolution of points in the Julia set
MaxIterations	Automatic	how many iterations to allow for each point
Method	Automatic	the method to generate the image
PerformanceGoal	$PerformanceGoal	aspects of performance to try to optimize
PlotLegends	None	legends for the number of interations
PlotRange	Automatic	range of values to include
PlotRangeClipping	True	whether to clip at the plot range
PlotStyle	Automatic	graphics directives to specify the style for each point
PlotTheme	$PlotTheme	theme to use for styles and appearances

The possible methods are "InverseIteration", "EscapeTime", and "OrbitDetection".
With Method->"InverseIteration", JuliaSetPlot produces a ListPlot showing individual points of the Julia set.
With Method->"EscapeTime" or with Method->"OrbitDetection", JuliaSetPlot produces a Graphics object containing a Raster primitive.
With ColorFunction->f, where f is a function, the argument of f is a real number in proportional to the number of iterates, and f must return color directives, such as RGBColor and Hue, or named colors, such as Red and Blue.
ColorFunction->"name" is equivalent to ColorFunction->(If[#1,Black,ColorData["name"][#]]&).
The list of possible color function names is given by ColorData["Gradients"].

Examples

open allclose all

Basic Examples (4)

Generate the filled Julia set of :

Generate the Julia set:

Show the filled Julia set with the points in the Julia set:

Show a legend of the number of iterations:

Scope (7)

JuliaSetPlot[c] generates the Julia set of a function of the form :

Generate the Julia set of a polynomial:

Generate the filled Julia set of a rational function:

Change the color function:

Show the points in the Julia set:

Show just the points:

Use a theme with simple ticks in a high-contrast color scheme:

Options (13)

Background (1)

Use a black background as contrast with the points in the Julia set:

ColorFunction (3)

Use a built-in color function to color by a scaled number of iterations:

Zoom in to a region to see more details:

Use a named color gradient:

Write a custom color function:

ImageResolution (1)

Increase ImageResolution for finer plots:

PerformanceGoal (2)

Generate a higher-quality plot:

Emphasize performance, possibly at the cost of quality:

PlotLegends (1)

Legends show the number of iterations:

PlotRange (2)

The plot range is chosen automatically, based on the Julia set:

Zoom in to a region to see more details:

PlotStyle (2)

By default, the points are not shown unless ColorFunction->None:

Specify the plot style:

PlotTheme (1)

Use a theme with increased contrast and axes:

Possible Issues (4)

JuliaSetPlot only works with rational functions:

When using orbit detection, sometimes no attractive orbits are detected:

If the value of the "Bound" method option is too low for a rational function, no points may be returned:

Some very large Julia sets can take a long time to compute:

Neat Examples (1)

Colorize the components of a Julia set:

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JuliaSetPlot

Details and Options

Examples

Basic Examples (4)

Scope (7)