Fakultät II
Institut für Mathematik

# Virtual Math Labs: Dynamical Systems

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Contents:
 KP2 Equation - Shallow Water Waves Pendulum Three models for the pendulum Planar Elastic String mKdV Spectral Curve Iterated Darboux Planar Vortices Smoke Ring Flow Top Visualization K-surfaces with a planar strip K-surfaces with a cone point

To start a lab just click on the screenshot. If the application does not start, have a look at our Help page.

### KP2 Equation - Shallow Water Waves

Markus Schmies This Application allows to explore solutions of the equation

4uxt + 3(u2)xx ± uxxxx - 3uyy = 0

discovered by Kadomtsev & Petviashvili (KP). The equation is a generalization of the Korteweg & de Vries (KdV) equation. Solutions of the KP2 equation describe the evolution of gravity-induced waves of moderate amplitude on shallow water of uniform depth when the waves are nearly one-dimensional. Features
• interactive graphical manupulation of the underlaying riemann surface
• interactive change of the genus
• interactive 3D viewer
• animation of the waves

### Pendulum

This application allows to explore the evolution of a chain of elastically coupled pendulums. Mathematically, it provides some insight into the dynamics of the Sine-Gordon equation.

Author: Ulrich Pinkall

### Three models for the pendulum

This lab allows to compare three modells for the pendulum.

rk: solves the pendulum equation numerically with the Runge-Kutta method.
sin: is a discrete symplectic model related to the standard map of chaos theory.
ddg: is a discrete integrable symplectic model related to discrete surfaces of constant Gaussian curvature known from discrete differential geometry.

Lab author: Ulrich Pinkall

### Planar Elastic String

This application allows to explore the evolution of an elastic string moving in the plane. Mathematically, it provides some insight into the dynamics of the wave equation.

Press 'e' to encompass and CTRL-Click to change between different transformation modes. For more information see the online help of the application.

Author: Ulrich Pinkall

### mKdV

This lab implents the discrete modified Korteweg-de Vries equation. Drag the larger yellow points to change the initial curve. Then click the play button.

Lab author: Ulrich Pinkall

Ulrich Pinkall

### Iterated Darboux

In this lab you may investigate the discrete Darboux transformation of a planar polygonal curve. The four vertices of all quadrilaterals in the evolution have fixed cross ratio. You may change the value of their cross ration on the right, where the yellow point is the cross ratio in the complex plane.

Lab author: Ulrich Pinkall

### Planar Vortices

In this lab one may investigate the planar vortex flow. Some suggestions for experiments:
Random configurations:
• press the play button
• press "e", scale or move the image by dragging with middle or left mouse to follow the moving vortices
• press the "Reset" button to start with a new configuration
More ...

Lab author: Ulrich Pinkall

Ulrich Pinkall

### Top Visualization

Program for the 3D visualization of a rigid body motion around a fixed point in a homogeneous gravity field at the example of quader. It includes the construction of the special cases of Kowalewski and Goryachev-Chaplygin, displays their integrals of motion and visualizes their corresponding elastic rods. Further it includes the possibility to compare different discretizations of the paths of lagrange tops, i. e. the integrable discretization developed in [1] and a numerical one.

For a complete description of content, usage and functionality see [2].
Besides the java web start there is a zip archiv of a stand alone version (Windows or Linux) of the program

#### Literature

[1] A. I. Bobenko and Yu- B. Suris, A Discrete Time Lagrange Top and Discrete Elastic Curves, Amer. Math. Soc. Transl. (2) Vol. 201, 2000.

[2] René Bodack, Dynamics of the Spinning Top: Discretization and Interactive Visualization, Diploma Thesis, unpublished (online).

### K-surfaces with a planar strip

A K-Surface is a surface of constant negative Gaussian curvature. The planar strip may be edited on the left. On the right you may investigate the surface and its Gauss map. The Gauss map may be seen as the evolution of massive balls on the sphere connected by rubber bands. There is a small mpeg (2MiB) video or animated gif (11MiB). More ...

For the theoretical background consult Designing Cylinders with Constant Negative Curvature.

### K-surfaces with a cone point

A K-Surface is a surface of constant negative Gaussian curvature. The initial Gauss map may be edited on the left. On the right hand side you may investigate the surface and its Gauss map. The Gauss map may be seen as the evolution of massive balls on the sphere connected by rubber bands. More ...

For the theoretical background consult Designing Cylinders with Constant Negative Curvature.

 Ulrich Pinkall . 12.10.2011.