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
The normalized power spectrum is calculated from the results of the Iteration. Note that only the first 2k points are used in the calculation of the Power spectrum, where k is the largest integer for which 2k is smaller than, or equal to N. The result below is for 256 iterations of the logistic map starting with initial values F1=0.0, F2=0.2 and a = 0.99. At this high value of the constant a the behavior is quite chaotic. This can also be seen in the bifurcation diagram. Note that this diagram shows a gap at a=0.96 where we find a 3-cycle. Indeed the power spectrum graph gives a nice peak at J=171 which means that the 256 iterations show about 85 ( =(171-1)/2 ) cycles.