Modes and Options
The system being graphed is the one with the current tab open. So if Complex Plane is open then the dynamical system uses the z = ... equation and it is graphed on the complex plane. If the Real Plane tab is open then it will graph x = ... and y = ... parametric equations. The syntax and available functions for the program are on the Expression Syntax page.
The Complex Plane tab has two modes of operation, Fix z and Fix c. In the Fix z mode the initial z value is given on that line and used in the first iteration of the dynamical system, z = .... After that, the subsequent values of z are fed back into the system. The value of c is taken from each point (pixel) in the complex plane image and is held constant through the dynamical system for that point. This can be thought of as Mandelbrot mode for the system since with this option and the formula z = z^2 + c, one will get the Mandelbrot set.
In the Fix c mode the value of c is held constant for each point (pixel) in the complex plane and the initial value of z is taken from each point (pixel) in the complex plane image. This can be thought of as Julia set mode for the system since with this option and the formula z = z^2 + c, one will get the Julia set associated with the complex number c.
The Real Plane tab has the equivalent options for the real plane. The Fix (x, y) mode is a Mandelbrot mode with the initial value of (x, y) taken from the panel and the (c, d) values range over the plane. The Fix (c, d) mode is a Julia set mode where the values of (c, d) are fixed throughout the generation of the image and the initial values of (x, y) are taken from the plane.
The other options apply to both types of systems.
- The image position is determined by the x and y values of the center. So for complex plane systems the center is x + yi and for real plane systems the center is (x, y). When you use the mouse in zooming or centering mode these values will automatically be updated by the mouse click and the mouse click will re-render the image.
- The width is the horizontal length of the viewing window. The height is automatically determined by the width and the aspect ratio of the current image.
- The maximum iteration is the largest number of iterations of the dynamical system, any point that make it through this many iterations is considered inside the set and those who hit the bailout radius before that are considered to be outside the set.
- The bailout radius is the radius used to determine if another iteration needs to be done. The bailout radius will change depending on the formulas being used. The default value of 4.0 is a point of no return for the Mandelbrot set and its associated Julia sets, but in general this value will need to be altered. In addition, some coloring schemes will work better if a larger bailout radius is used. If the bailout mode is set as escape then an orbit bails out if the modulus of a point in the orbit exceeds the radius. If the bailout mode is set as convergence then the orbit bails out if the distance between two consecutive points in the orbit is smaller than the radius.
- The bailout mode is either escape or convergence. In escape mode an orbit bails out if the modulus of a point in the orbit exceeds the radius. In convergence mode then the orbit bails out if the distance between two consecutive points in the orbit is smaller than the radius. Escape mode should be used on systems similar to the Mandelbrot set and Julia sets and convergence mode should be used on systems similar to Newton's Method systems.
- The plane is either normal complex or (x, y) plane and the inverse plane is the 1/z plane where each point z is replaced by 1/z.