This is an archived page from the year 1997. Olivers page is here.

Gif Movies of Oliver Knill

The Standardmap (22K) (x,y) -> (2x+c sin(x)-y,x) (mod 1), for c=0.0...3.0 . The colors are according to the Lyapunov exponent.
The billiard map (66K) with table |x|^p + |y|^p = 1 for p =1 ...4 . We see the phase space for each p. The cases p=1,2,infinity are integrable, the others not. We measured positive Lyapunov exponents for other p. The data to the picture here were produced by Monwhea Jeng.
The Cubic Henon family (157K) (x,y) -> (cx -x^3 -y,x), for c=2 .. 3. The colors are according to the time, an orbit stays in a specific bounded region. We see a bifurcation from a fixed point into two fixed points.
The game of life (88K) with almost periodic initial conditions. Every picture is part of an infinite aperiodic phase space. The pictures wer generated using the Mathematica routines in joint work with Bert Hof .

The Lorentz system (66K) is maybe the most famous differential equation. We fly around the Lorentz attractor.
Julia sets (52K) of the quadratic map z -> z*z+c. The picture shows, how the Julia set changes when the parameter moves on the unit circle in the c-parameter plane.
Some caustics (39K) of a family of strictly convex billiards.
Dynamics on Sets (32K) in the Standard map illustrates the mixing property of the map. Until now, it is open, whether there is a set of positive measure, on which the dynamics is really mixing. A longer iteration (94K) of the same map.

Vortices in the plane (17K) A vortex street on the circle is unstable (an old result of Thomson). Here we see 31 vortices inicially alligned on the circle. Integrated with Runge-Kutta with a Mathematica code allowing to evolve an arbitrary number of vortices.
The Standard Legendrian collaps (250K) A wave front moving inside a surface of equal thickness is collapsing, turning inside out and radiating away. The place, where the smoothness of this process gets lost is the caustic of the surface.
Standard Legendrian collaps for a billiard (30K) A wave front radiating from a curve of equal thickness collapses, turns inside out and radiates away. The place, where the wavefront fails to be an embedding is the caustic of the curve.
The double pendulum (28K) integrated with Runge Kutta is a Hmiltonian system which behaves chaotically in certain regimes. It is an open unsolved question whether this system has positive Lyapunov exponents on a set of positive measure.

Flight over the Henon Attractor There is a one line Mathematica program producing the attractor: T[{x_,y_}]:={0.3*y+1-1.4*x^2,x};ListPlot[NestList[T,{.56,.37},10^4]]
A plasma on the circle (100 frames) moving under a Vlasov dynamics. The initially smooth curve in the phase space becomes more and more wiggly. The growth rate of the gradient defines a Lyapunov exponent for the infinite dimensional Vlasov system.
2 particles on the circle (300 frames) The actually integrable system becomes unstable by the time discretisation due to a singular potential (the natural Newton potential on the circle is V(x)=|x(1-x)|).

The second longest Gif movie (repeats only after 2000 current ages of the universe).

The longest Gif movie (repeats after more than 1 googol (10 ¹⁰⁰) ages of the universe). Has more frames than the number of atoms in the universe.

These Examples were done using MATHEMATICA on a SUN or my NEXT, converted to GIF files under UNIX using the PBM routines and then processed with GIFBUILDER 0.3 (a program of Yves Piguet. It is downloadable here ) on my Powerbook MAC. Recent versions were done with my Mathematica Package MATHEMATICA GIF movies under UNIX which allows Mathematica directly to export the Gif Movie. Here my Mathematica Package GMT_3.0.m for Gif-Movie exportation with Mathematica 3.0 You might need the gifmerge.tgz package and the netpbm.tgz package.

Beside my Mathematica code for the movies, I used data obtained by Monwhea Jeng (using a C code written during a SURF project at Caltech ). The almost periodic Cellular Automata Movie uses Mathematica CA routines developed together with Bert Hof while doing research on almost periodic CA.

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