Approximation of continuous dynamical systems by discrete systems
	Introduction Installation Applets More Editorial

A comprehensive guide to continuous and discrete dynamical systems, equipped with an interactive laboratory to teach/learn the analysis and the approximation of solution of differential equations - submitted by Patrizia Nardin, Holler -

Introduction

   This job is motivated by the integration of Dynamical System Perspective and by the Graphical and Numerical Philosophy, which are developing in the mathematical ambit. The attention is focused on numerical approximation and a special investigation is made on long term analysis, which is a rather new investigation field.
   Play around, looking what keeps your interest and jumping through the sites, when somewhat makes up the curiosity, then you can go into the mathematical treatment ( which will be a reference on understanding the background of the applets).

Installation for Windows 9x/NT/2000

   You must have a Java 2-Plug-in (preferred: Java 1.3-Plug-in) installed.
Otherwise:
   Download and install the Java 1.3-Plug-in (file: jre1_3-win.exe) in your favorite directory (e.g. directory: C:\ProgramFiles\JavaSoft\JRE\1.3).
For any troubleshooting information, check out the Java FAQ.

Applets

   Three famous continuous dynamical systems are considered: the Dahlquist test, the Verhulst equation and the Lotka-Volterra system (competing species model).

Dahlquist test
	Vector Field Through the geometrical representation of the slope function, an intuition of the system's solution is obtained.
	Euler Method Let us see the first studied numerical method in order to understand how an approximation of the exact solution may be produced.
	Heun Method This numerical method let us introduce the concept of accuracy. You may test the practical significance.
	RungeKutta Method This is the most applied method. The geometric representation of the method gives an elucidation of the numerical technique.
	LongTerm Behaviour After a large interval of time how asymptotic is the numerical approximation to the analitical solution?

Verhulst equation
	Vector Field Sensitivity to initial-value is a typical feature of a non-linear dynamical system.
	Euler Method The caos may appear if you will be uncareful applying the method.
	Heun Method You may graphically test the range of the stability condition of the method, but analytically?
	RungeKutta Method You have now the ability to bear a comparison of method's results! An additional step is the extension of the Ratio Study to non-linear systems.

Lotka-Volterra system
	Vector Field The refinement of the vector field representation is of importance in order to supply a good guide in studying the qualitative behaviour.
	Numerical Approximation In comparing the method, you can make as well some considerations on the iteration's computational effort.

Do you want to know more ...

• about Dynamical Systems ?

You get a lot of information about this field in Internet, there are some cues.
 
• about the Mathematical Treatment?

Take a look at the complete thesis.

Editorial

Location of this file: http://www.gris.uni-tuebingen.de/projects/dynsys/index.html