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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 Julia sets are produced when stable
orbits are searched in the plane of initial conditions (F3-F4 plane)
while keeping a and b fixed. You can give the a and b values in the
Pixel window below the graph. Confirm the new choice of a and b by pushing
the Reset constant values button below
the value entries. Alternatively you may select a value in the Mandelbrot
set by pushing the Select coordinate button and use you mouse (left button)
to select the coordinate in the graph.

The pictures below were produced in this way.
Exscape (black region) for F5>2, maximun number of iterations 100