## Iterations

## 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.

**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

## Mandelbrot sets – Stable orbits

The Mandelbrot sets are produced when stable orbits are searched in the a-b plane. This is done for the initial values F3=F4=0 (z=0) (F5 is used in the stability criterium)

It was made in the following steps:

Choose the Iterations module and Push the *Prepare / Draw* button to open the Prepare Iterations window. Push the *Iteration type* tab ans choose *Vary two parameters or initial values*. Here you have to set the maximum number of iterations and the maximum value of the escape parameter (F5 in this case). Note that all 5 functions (F1, F2, ..F5) in the Functions panel have to be selected. Choose *Vary parameters* (X=a, Y=b) and *Open the Pixel Graph* window. Set the range of a and b in this window and push the *Iterate and Draw* button. In the picture below the maximum number of iterations was 50 with F5< 4 (Black region).

The first picture below is an enlargement of the region indicated above. You may zoom in by pushing the Select new range button and click on the graph (left button) to give the new lower left and upper right corners. In the pictures below the max. number of iterations was 75, 100, 200, 200 and 600 resp. (ordered clockwise). The color may be adjusted in the panel below the graph. Not how the final structure resembles the original structure.