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
Henon map – Stability
The picture below shows the region of stability, i.e. the region from which orbits do not escape. It was produced as follows:
Select the first 5 Functions in the Function panel. The fifth function (F5) gives the distance to the origin. It is used in the escape criterium. Go to Iteration type in the Prepare / Draw graph window and choose Vary two parameters or initial values. Choose Vary initial values. Set Set the maximal number of iterations to 50 and the Escape value of F5 to 10. Open the Pixel grpah, set the minimum and maximum coordinates to -2 and 2 respectively, and push the Iterate and Draw button.
The red orbit was added in a next step by going to Standard type Iteration, choosing Draw results directly … and push Iterate and Draw
The picture below shows the stable orbits of the Henon map for a=0.2 , b=1.1 in the region F3= -4 to 4 and F4 = -6 to 6. The maximum number of iterations was set to 100 and the escape: F5>10 (black region). The colors give the number of iterations it took to escape. The color range can be set in the lower part of the graph window. Here they go from red to blue (rainbow mode).