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
Logistic map: Sensitivity to initial conditions
The logistic map is perhaps one of the simplest mathematical system showing many characteristics of the development of chaotic behavior. It is a simple iterative map which you can readily program on your calculator:
This simple map, which was extensively studied by Feigenbaum, shows very intriguing behaviour when the value of a is increased. For a smaller than 0.75 the x-value quickly converges to a single value (fixed point). As a is increased above 3 the value of x alternates between two values (2-cycle). As a is increased further the period is doubled: it goes into a 4-cycle. Such period doublings keep occurring when a is increased further until the behavior becomes completely chaotic. Feigenbaum found that this type of behavior – the development of chaos – occurs in many systems in a similar way. The period doubling can be seen in the “bifurcation diagram”. Start the demonstration in Mathgrapher to see how it is produced. It contains the final values of x for a series of iterations with increasing a. It shows how the system evolves from fixed point through 2-, 4- and 8- cycle (and 3-cycle at a=0.96) into chaotic behavior.
Below the result is shown of 30 iterations of the logistic map ( File=>open=>Function=>Maps=>logistic.fct) for a=0.94 After defining and selecting F1 and F2 in the
Iterations panel, the Prepare / Draw window is opened, the initial values of F1 and F2 are set in the Start / End conditions panel. Push the Iterate button. Finally select Time Series in the combo box on the Analysis panel, select F1 and F2 and push Draw to produce the Graph below. The red line is the result of setting the
initial values F1=0.0 and F2=0.2. The black line starts at F1=0.0 and F2=0.202. Note how a small variation in initial values causes a sudden change after 14 iterations.