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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 logistic map is perhaps one of the simples mathematical system showing
many characteristics of the development of chaotic behaviour. It is a
simple iterative map which you can readily program on your calculator:

This simple map, which was extensively studied by Feigenbaum, shows very
intriguing behaviour when the value of a is increased. For a smaller than 0.75
the x-value
quickly converges to a single value (fixed point). As a is increased above 3 the
value of x alternates between two values (2-cycle). As a is increased
further the period is doubled: it goes into a 4-cycle. Such period doublings
keep occurring when a is increased futher until the behaviour becomes
completely chaotic. Feigenbaum found that this type of behaviour - the
development of chaos - occurs in many systems in a similar way. The period
doubling can be seen in the "bifurcation diagram". Start the demonstration
in Mathgrapher to see how it is produced. It contains the final values of x
for a series of iterations with increasing a. It shows how the system
evolves from fixed point through 2-, 4- and 8- cycle (and 3-cycle at a=0.96)
into chaotic behaviour.

Below the result is shown of 30 iterations of the logistic map
( File=>open=>Function=>Maps=>logistic.fct) for a=0.94 After defining and selecting F1 and F2 in the
Iterations panel, the Prepare / Draw window is opened, the initial values of F1 and F2 are set in the Start / End
conditions panel. Push the Iterate button. Finally select Time Series in the combo box on the Analysis panel,
select F1 and F2 and push Draw to produce the Graph below. The red line is the result of setting the
initial values F1=0.0 and F2=0.2. The black line starts at F1=0.0 and F2=0.202.
Note how a small variation in initial values causes a sudden change after 14 iterations.