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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 Fifth function F5 (distance to the center) is used
in the stability criterium (you may import this map by going to the main menu:
File => Open=> Function. You find Mandelbrot.fct in the Maps subdirectory).
The colorful fractal pictures of the Mandelbrot set are made by doing iterating the equations for each coordinate (a,b). The pixel colors are
determined by the number of iterations it takes until the value of |z| =sqrt(x^2+y^2) exceeds a certain value. The Julia set for c (a,b) is a set
of initial values z (x,y) -values for which the |z| does not go to infinity when iterated. Julia sets have the interesting property that they are connected
for points (a,b) inside the Mandelbrot set and disconnected for points lying outside the set.