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Iterations

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 simples mathematical system showing many characteristics
of the development of
chaotic
behaviour. Several analytical tools
are valailable to study the results of Iterations and ODE's such as:

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.

There are several types of fixed points depending on the behaviour
of the orbits in their neighbourhood: stable or elliptic, hyperbolic, parabolic The behaviour
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.