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

Julia sets – Definition

The Julia sets are produced when stable orbits are searched in the plane of initial conditions (F3-F4 plane) while keeping a and b fixed. You can give the a and b values in the Pixel window below the graph. Confirm the new choice of a and b by pushing the Reset constant values button below the value entries. Alternatively you may select a value in the Mandelbrot set by pushing the Select coordinate button and use you mouse (left button) to select the coordinate in the graph.

The pictures below were produced in this way. Exscape (black region) for F5>2, maximum number of iterations 100