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# Modules

Iterations
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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:
 time series | Power spectra | 2D projections | Fixed points | Lyapunov exponents
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.

Examples:
 Logistic map: Sensitivity to initial conditions Projection in 2D Power spectrum Bifurcation diagram Lyapunov exponents Henon map: Definition 2D orbit Region of Stability Mandelbrot and Julia sets: Definition Mandelbrot: vary parameters Julia: vary initial conditions

# Bifurcation diagram

applied to the logistic map:

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).

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