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.
Maps - Lyapunov exponents - Definition and calculation.
Lyapunov exponents are very important and useful in
the description of chaotical dynamical systems. They describe the exponential rate
at which neighbouring orbits diverge. They can be used to determine the behaviour
of quasi-periodic and chaotic behavious as well as the stability of equilibrium points
and periodic solutions. For orbits near equilibrium points they are equal to the
real parts of the eigenvalues at these points.
An n-dimensional map has n Lyapunov exponents. Mathgrapher calculates all of them.
The method used is the one described in
"Practical Numerical Algorithms for Chaotic Systems"
by T.S. Parker and L.O. Chua (1989, Springer-Verlag New York Inc.).
The calculation of the exponents is easier than in the case of ODE's.
of the map at each point x(i) along the orbit, i.e. calculating the tangent
map M(i), For an n-dimensional system this is a n x n matrix. The tangent
map gives the amount of expansion or contraction of neighbouring orbits near xi.
After a few iterations the product M(k)M(k-1) ..M(1) is likely to grow rapidly
and become ill-conditioned. So we have to apply orthonormalization at regular
intervals as in the case of ODE's. This is done by QR decomposition.
The lyapunov exponents are calculated from the diagonal elements of
R (see also ODE's).
The simplest case is the one-dimensional map
(see for example the
For n-dimensional maps
For area conserving maps contraction in one direction is
balanced by expansion in another direction, so we have