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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.

Below the result is shown of iterates of the logistic map for a ranging from a=0.72 to a=0.98
(see also Demonstrations=>Iterations=>Logistic map). Note the period doublings at a=3/4 and 0.862 and higher.
As a increases the behaviour becomes increasingly chaotic especially above the gap at a=0.96 where
a 3-cycle occurs (see the power spectrum).