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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.

Thenormalised power spectrum is calculated from the results of the Iteration. Note that only the first
2**k points are used in the calculation of the Power spectrum, where k is the largest
integer for which 2**k is smaller than, or equal to N.

The result below is for 256 iterations of the logistic map starting with initial values F1=0.0, F2=0.2 and a = 0.99.
At this high value of the constant a the behaviour is quite chaotic. This can also be seen in the
bifurcation diagram.
Note that this diagram shows a gap at a=0.96 where we find a 3-cycle. Indeed the power spectrum graph
gives a nice peak at J=171 which means that the 256 iterations show about 85 ( =(171-1)/2 ) cycles.