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Iterations

Calculate
algebraic series
such as e = 1+ 1/2! + 1/3! + ..., a square wave,Fibonacci numbers.
Study iterative maps, e.g. the (one-dimensional)
Logistic map:
(see below) or more complicated multi-dimensional maps. The logistic map is perhaps one of the simples mathematical system showing many characteristics
of the development of
chaotic
behaviour. Several analytical tools
are valailable to study the results of Iterations and ODE's such as:

A special window has been added in Mathgrapher v2 to allow detailed presentation
of 2D orbits at the pixel level and to study the stability of the orbits (see the Examples and Demonstrations for the Henon map, Standard map and Mandelbrot and Julia sets.

The Mandelbrot sets are produced when stable orbits are
searched in the a-b plane. This is done for the initial values F3=F4=0 (z=0)
(F5 is used in the stability criterium)
It was made in the following steps:
Choose the Iterations module and Push the
Prepare / Draw button to open the Prepare Iterations window.
Push the Iteration type tab ans choose
Vary two parameters or initial values.
Here you have to set the maximum number of iterations and the maximum value
of the escape parameter (F5 in this case). Note that all 5 functions (F1, F2, ..F5)
in the Functions panel have to be selected. Choose
Vary parameters (X=a, Y=b) and Open the Pixel Graph
window. Set the range of a and b in this window and push the
Iterate and Draw button. In the picture below the maximum number of
iterations was 50 with F5< 4 (Black region).

The first picture below is an enlargement of the
region indicated above. You may zoom in by pushing the Select new range button
and click on the graph (left button) to give the new lower left and upper right
corners. In the pictures below the max. number of iterations was 75, 100, 200,
200 and 600 resp. (ordered clockwise). The color may be adjusted in the panel
below the graph. Not how the final structure resembles the original
structure.