## 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 simplest mathematical system showing many characteristics of the development of chaotic behavior.
Several analytical tools are valuable 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.

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 behavior becomes increasingly chaotic especially above the gap at a=0.96 where
a 3-cycle occurs (see the power spectrum).