## 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 map – Definition

The Mandelbrot set is the set of points in the complex c-plane for which |z| does not go to infinity when iterating

starting with z = 0. Defining z=x+iy and c=a+ib this yields the 2D map:

In Mathgrapher this looks like:

F1=F3

F2=F4

F3=F1^2-F2^2 +a

F4=2F1F2 +b

F5=sqrt(F1^2+F2^2)

The Fifth function F5 (distance to the center) is used in the stability criterium (you may import this map by going to the main menu: File => Open=> Function. You find Mandelbrot.fct in the Maps subdirectory). The colorful fractal pictures of the Mandelbrot set are made by doing iterating the equations for each coordinate (a,b). The pixel colors are determined by the number of iterations it takes until the value of |z| =sqrt(x^2+y^2) exceeds a certain value. The Julia set for c (a,b) is a set of initial values z (x,y) -values for which the |z| does not go to infinity when iterated. Julia sets have the interesting property that they are connected for points (a,b) inside the Mandelbrot set and disconnected for points lying outside the set.