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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 pictures below show some orbits for the Henon map with a=1.4, b=0.7.
The first one is made in the following steps:
First select functions F1 to F4 in the Function Panel.
Next open the Prepare / Draw Iterations window => open the Iteration type tab.
We choose to Draw results (2D) directly in the Pixel Graph window.
Next set the number of iterations to 3000 and the initial coordinates
F3=0.7 and F4=0.3. Push the button that opens the Pixel Graph window,
set the number of pixels at 300x300 pixels, set the X coordinate (F3)
from -2 to 2, the Y-coordinate (F3) from -2 to 2 and push the
Iterate and Draw button in the Prepare / Draw iterations window. This will
yield the first picture on the left (apart from the square).
The square results from
selecting a new range. The middle and right pictures are made by selecting a new (smaller) range and
doing a new iteration. The middle picture below was obtained for 30.000
iterations and the picture to the right for 3 runs of 1000.000 iterations
(3 different initial positions on the attractor). Note how the fine fractal (cantor-like) structure of the orbit becomes visible.

The picture below shows orbits of the Henon map
for a=0.2 , b=1.1 in the region F3= -4 to 4 and F4 = -6 to 6. You may use
your mouse to select new initial values from the graph after pushing the button above the Graph.

It is also interesting to compare this graph with a graph that shows where the most
stable orbits are