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
Consider a 2-dimensional map T given by
The map has a fixed point X when
There are several types of fixed points depending on the behaviour of the orbits in their neighborhood: stable or elliptic, hyperbolic, parabolic The behavior of the orbits near a fixed point can be found by studying the linearized equations for small perturbations around the fixed point.
The eigenvalues of the matrix T determine the type of equilibrium point and its stability properties. They are the roots of the equation
where I is the identity matrix. In the 2-dimensional case discussed here this generally yields 2 solutions. The eigenvalues are sometimes called characteristic multipliers. Just like the eigenvalues at an equilibrium point their position in the complex plane determines the stability near the fixed point. The magnitude of the eigenvalues give the amount of contraction or expansion near the fixed point and must herefore be equal to the Lyapunov exponents near that point. Below are some orbits drawn for the Henon map
for a=0.2 and b=0.998
The two fixed points are at (2.23,2.23) and (-2.23, -2.23). The eigenvalues in the first fixed point are -1.54 and 0.647. The eigenvalues in the other point have opposite signs.