## 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

## Projection in 2D

Applied to the logistic map:

Choose a pair of F’s from the Functions that are iterated and push Draw to make a graph where the X-axis represents the first and the Y-axis the second Function.
The example below is a graph of F1 and F2 for a series of 1024 iterations of the logistic map. The first 100 iterations are omitted. Note that a 4-cycle has developed for a=0.87.

Same as above for a=0.89 showing an 8-cycle and a=0.96 (the gap in the bifurcation diagram) where we have a 3-cycle. The behavior is chaotic above 0.96.