Chaotic self-organizing behavior is an interesting phenomenon that can be seen in some instances of higher-dimensional Lotka-Volterra Equations. An excellent article on the conditions in which it occurs, and an example of such a system can be found here:[br][br][url=https://sprott.physics.wisc.edu/pubs/paper288.htm]https://sprott.physics.wisc.edu/pubs/paper288.htm[/url][br][br]We won't have a chance to study this topic in this course, however, to whet your appetite for this interesting area of study, the example of a 4 dimensional Lotka-Volterra System, and Figure 3 from the above article has been replicated in the downloadable .GGB (GeoGebra) file for you to explore. A screenshot of the file is below:

This is an animated GIF of the same file.

You can download the .GGB (GeoGebra) file here[br][br][url=https://cloudup.com/cH0HS72m6RB]https://cloudup.com/cH0HS72m6RB[/url][br][br]You will need GeoGebra Classic 5 or 6 Desktop to open this file. [br][br][i]Note[/i]: The .GGB file may take up to 5 minutes to load on your computer, but once it's started it should work quickly. Also, after you wait for the initial launch, you may need to "jiggle" [code]InitialCondition[/code] to jump start the numerical ODE solver.[br][br]In the file you will find a 3-dimensional projection of the phase space perspective of the specific solutions of the 4-dimensional Lotka-Volterra System studied in the above article. [br][br]Chaotic self-organizing behavior can be observed in the file by adjusting the point [code]InitialCondition[/code]. When you do so, observe that small changes in [code]InitialCondition[/code] result in large differences in the specific solutions. Despite these large differences in the specific solutions, however, note that the solutions nonetheless exhibit self-organizing behavior oscillating around a certain point in the first octant.[br][br]If you REALLY want to try loading it in your web-browser, you can try doing so here: https://www.geogebra.org/m/sscerf4m But be warned, it may take a long time to load in browser.[br]