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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 picture below shows the region of stability, i.e.
the region from which orbits do not escape. It was produced as follows:
Select the first 5 Functions in the Function panel. The fifth function
(F5) gives the distance to the origin. It is used in the escape criterium.
Go to Iteration type in the
Prepare / Draw graph window and choose Vary two parameters
or initial values. Choose Vary initial values.
Set Set the maximal number of iterations to 50 and the Escape value of F5 to 10.
Open the Pixel grpah, set the minimum and maximum coordinates to -2 and 2 respectively,
and push the Iterate and Draw button.
The red orbit was added in a next step by going to Standard type
Iteration, choosing Draw results directly ...
and push Iterate and Draw.

The picture below shows the stable orbits of the Henon map
for a=0.2 , b=1.1 in the region F3= -4 to 4 and F4 = -6 to 6. The maximum number
of iterations was set to 100 and the escape: F5>10 (black region). The colors
give the number of iterations it took to escape. The color range can be set in the
lower part of the graph window. Here they go from red to blue (rainbow mode).