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 – Orbits
The pictures below show some orbits for the Henon map with a=1.4, b=0.7. The first one is made in the following steps:
First select functions F1 to F4 in the Function Panel. Next open the Prepare / Draw Iterations window => open the Iteration type tab. We choose to Draw results (2D) directly in the Pixel Graph window. Next set the number of iterations to 3000 and the initial coordinates F3=0.7 and F4=0.3. Push the button that opens the Pixel Graph window, set the number of pixels at 300×300 pixels, set the X coordinate (F3) from -2 to 2, the Y-coordinate (F3) from -2 to 2 and push the Iterate and Draw button in the Prepare / Draw iterations window. This will yield the first picture on the left (apart from the square).
The square results from selecting a new range. The middle and right pictures are made by selecting a new (smaller) range and doing a new iteration. The middle picture below was obtained for 30.000 iterations and the picture to the right for 3 runs of 1000.000 iterations (3 different initial positions on the attractor). Note how the fine fractal (cantor-like) structure of the orbit becomes visible.
The picture below shows orbits of the Henon map for a=0.2 , b=1.1 in the region F3= -4 to 4 and F4 = -6 to 6. You may use your mouse to select new initial values from the graph after pushing the button above the Graph.
It is also interesting to compare this graph with a graph that shows where the most stable orbits are