Visualization of a 4 dimensional dynamical system

The Dynamical System

The dynamical system we are going to investigate is definded by the following differantial equations:

pi1`=-12pi4+2e(m-1)pi1

pi2`=12pi4*2e(m-4)pi2

pi3`=6pi4+3e(m-2)pi3

pi4'=-6pi3-12pi1*pi2+6pi1^2+3e(m-2)pi4

There are four system variables, pi1, pi2, pi3, pi4, which define the 4D phasespace and two system parameters: e and m. The e parameter will be assumed constant e=0.1 throughout the visualization, while m will be swept from -1 to 5 to visualize changes in the system´s behaviour.

The investigated system is a transformed two-mode approximation of the one dimensional perturbed Korteweg-de Vries equation (KdV) and has been used during a mathematical proof in [vGSo].

The KdV quation plays an important role in wave dynamics, describing extremely stable solitary waves called solitons. An application of solitons which might be of relevance to the reader accessing this pages through the WorldWideWait is fast data transmission with rates of 20 Gbit/s over 14000 kilometers.

Unfortunately, there is no intuitive interpretation of the system variables pi1 through pi4, like the meaning of the axes of the Lorenz system, which makes the interpretation of the system even more difficult.