simStudio

You can choose between three different nonlinear systems to be simulated:
 

A driven pendulum without damping
A kicked rotator
A billiard system with a cos-shaped border
In each simulation there is a large poincare section and sliderbars on the right hand side for system specific parameters.
If you press the right mouse button in the poincare section a popup window appears. The initial condition can be set by pressing the left mouse button in the poincare section.

Context menu functionality:

Clear:
Clears the Poincare section window
Zoom / Zoom back / Zoom back all:
It is possible to zoom into the Poincare section. When choosing Zoom, a zoombox appears. The position can be altered by moving the mouse, the size by pressing the left mousebutton and dragging.
It is also possible to rotate the zoombox by pressing the shift-key and moving the mouse. (The angle of rotation can't be larger than 45 degrees) If the zoombox has the desired size and amount of rotation press the right mousebutton to confirm.
"Zoom back" lets you zoom back one step, "Zoom back all" returns to the initial poincare section.
Cloud of initial conditions:
With this option you can inspect the evolution of a Gaussian distributed Cloud of initial conditions Click into the Poincare window and drag for a cloud of desired size.
Small / Medium / Large dots:
The dot size for points drawn in the Poincare section.
The Driven Pendulum


Literature:

Doerner R, Hubinger B, Heng H, Martienssen W. Approaching nonlinear dynamics by studying the motion of a pendulum. II. Analyzing chaotic motion. [Journal Paper] International Journal of Bifurcation & Chaos in Applied Sciences & Engineering, vol.4, no.4, Aug. 1994, pp.761-71. Singapore.

Heng H, Doerner R, Hubinger B, Martienssen W. Approaching nonlinear dynamics by studying the motion of a pendulum. I. Observing trajectories in state space. [Journal Paper] International Journal of Bifurcation & Chaos in Applied Sciences & Engineering, vol.4, no.4, Aug. 1994, pp.751-60. Singapore.


Parameters can be changed either by manuipulating the corresponding scroll bar or by typing into the number field. There are the following parameters:
    F: The Ampltude of the driving force.
    Omega: The driving angular velocity. 
Additional context menu functionality:
Show Pendulum shows a visualization of the pendulum.
The Kicked Rotator

The Kicked Rotator (also known as Chirikov standard map) is a simple time dependent dynamical system showing chaotic dynamics. It is defined by the iteration

where theta is an angle, and p the angular momentum. With the simulation below you can observe classically and quantum mechanically the transition from regular behaviour for small values of K to chaotic dynamics for larger K.

Initially you see the classical phase space with momentum p in the interval [0,2*pi] vs. angle theta in the interval [0,2*pi] with periodic boundary conditions in p and theta, respectively. You can follow the time evolution of either a single classical trajectory, a cloud of classical trajectories or a quantum mechanical wave packet.

Literature:

Izrailev FM. Simple models of quantum chaos: spectrum and eigenfunctions. [Journal Paper] Physics Reports, vol.196, no.5-6, Nov. 1990, pp.299-392. Netherlands.


Parameters can be changed either by manuipulating the corresponding scroll bar or by typing into the number field. There are the following parameters:
    K: Kicking strength (see equation above)
    M cells: Sets the interval in momentum to [0,2*pi*M]. Choose a large value to see diffusion.
When in Husimi mode: (see below):
N: Sets hbar = 2*pi/(2^N) for the quantum mechanical simulation. For larger N the classical phase space is resolved on a finer scale, but it takes longer to calculate the time evolution. 
Additional context menu functionality:
With a Husimi plot you can compare the evolution of a quantum mechanical wave packet with a cloud of classical trajectories. When you click the left mouse button a coherent state wave packet as well as a classical cloud of same size is started. (Hint: The smaller the parameter N, the faster the simulation runs!) 

Show Rotator visualizes the rotator.

The Cosine Billiard


Parameters can be changed either by manuipulating the corresponding scroll bar or by typing into the number field. There are the following parameter:
    w: The size of the left and right border (see figure)
    m: The height of the cosine shaped "topping"  (see figure)
Literature:
P. Stifter, Diploma thesis, Universitaet Ulm, 1993 (unpublished)
P. Stifter, Ph. D. thesis, Universitaet Ulm, 1996 (unpublished)
G. A. Luna-Acosta et al., Phys. Rev. B 54, 11410 (1996)
A. Baecker, R. Schubert and P. Stifter, J. Phys. A: Math. Gen. 30, 6783 (1997)
Additional context menu functionality:
You can switch off the billiard ball by choosing "Fast"..



Please Note:
The applet may run very unstable on certain browser implementations. This is due to implementation bugs. The class files itself may be download here, which perform much more stable with sun's appletviewer.  (Note: The class files have to be unpacked in a directory named "redesign") A more convenient way to circumvent hangups might be to stop the simulation before changing a parameter.

Comments and bug reports are very welcome!


Idea: Roland Ketzmerick, Bjoern Naundorf
Java programming: Bjoern Naundorf

Please send Comments and bug reports to Bjoern Naundorf
This page was last modified on Tuesday, June 29, 1999.