simStudio |
start simulator here
You can choose between three different nonlinear systems to be simulated:
A driven pendulumIn each simulation there is a large poincare section and sliderbars on the right hand side for system specific parameters.
A kicked rotator
A billiard system with a cos-shaped border
Context menu functionality:
Clear:Clears the Poincare section windowZoom / 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.Cloud of initial conditions:
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.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:
Omega: The driving angular velocity. 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:
M cells: Sets the interval in momentum to [0,2*pi*M]. Choose a large value to see diffusion. 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!) |
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:
m: The height of the cosine shaped "topping" (see figure) P. Stifter, Diploma thesis, Universitaet Ulm, 1993 (unpublished)Additional context menu functionality: You can switch off the billiard ball by choosing "Fast".. |
Comments and bug reports are very welcome!