Chaos

Chaos

Introduction

People never cease to amaze me. By that I mean both laymen and ‘scientists’. I’ve seen many opinions of laymen about some sort of a ‘balance’ existing in all sorts of systems that no knowledgeable individual could claim to be in equilibrium. I’ve seen pretenses that the evil and sinning humans are ‘disrupting’ that balance. Salvation comes of course either through ascetism and suffering or by paying indulgences in form of taxes. It’s very like in the old religions, things do not change really much in the cargo cult ones. No matter if it’s about individuals, in psychology or about groups of people in sociology, or about climate, or something from biology, you see this a lot. You see it combined, too. Psychology mixed with climatology, climatology paired with biology, sociology with climatology, you see all sorts of ‘predictions’ or post hoc ‘explanations’ of some change, usually the more catastrophist ones being selected by the media and pushed onto gullible individuals. As if the publishing bias and other issues from sciences are not enough, the publishing bias from mass media on top certainly helps… Those ‘sciences’ have many problems, for example they are vague enough and allow rationalizations being usually protected by weasel words that allow them to predict things that are contradicting each other without instantly being proved false.

Some theory

Anyway, this post is not about those issues, although the topic is related strongly with those. I wanted to have at least a post on this blog about chaos theory, I’ll present here a model that is related with many things I mentioned above. It’s about a model of population dynamics, more specifically, the Competitive Lotka–Volterra equations. You should see the connection by now, in biology they study the animals that are modeled with those equations as interacting, humans are also animals and the climate is defined by averaging all sorts of very complex, nonlinear things which include… biology. Now, this model presented here is very simple, chaos theory still fights with very simple models, while cargo cult sciences act like they can predict the very complex systems (without actually predicting them in the scientific sense). I’ll just let you watch some videos and give links to papers instead of writing a lot here. The associated program1 is simple enough, too. It’s just an application of the Runge–Kutta methods together with a four-dimensional visualization with VTK (a 3D chart together with the color for the fourth dimension).

First, here is the program in action:

You can see that the system exhibits a nice strange attractor.

Second, here is the main article that you should consult: Chaos in low-dimensional Lotka–Volterra models of competition by JA Vano, JC Wildenberg, MB Anderson, JK Noel and JC Sprott2. Here is a lecture that might help: Lotka-Volterra Dynamics – An introduction by Steve Baigent3.

An easy to follow presentation on youtube:

If that opened your appetite, here are the publications of Edward Lorenz4.

Since I mentioned climate, too, here is a nice lecture about a toy climate:

If you failed to visit the Lorenz papers link4 and you believe that the climate is not weather I would suggest you to visit this link from the Church itself. Funny how they talk about ‘balances’ there, isn’t it? Anyway, it’s from an older report, they swiped that under the rug in the newer ones. It’s not that the climate suddenly became non chaotic, though.

The code

I tested the Runge-Kutta code from the Electric Field Lines project, changing the code slightly (the change is also in the original project, to keep the code in sync), and I displayed the results with VTK5. The application is also using wxWidgets6.
Besides the Runge-Kutta implementation, you might want to look into CompLKFunc.h to see how the equations are implemented. To see how they are used together with the Runge-Kutta solvers, check out CompLKFrame::Compute (both of them). If you are interested in the visualization, all CompLKFrame class might be of interest.

Conclusion

I could have something more interesting perhaps, some system like a double-pendulum or even more complex, starting from two very close points, animated, but at least for the double pendulum you can find tons of those, even on youtube.
Here is an example:

I’ll let that for some other time. As usual, if you find mistakes in the code or you have something to add, please do so.


  1. Competitive Lotka–Volterra The project on GitHub 
  2. Chaos in low-dimensional Lotka–Volterra models of competition by JA Vano, JC Wildenberg, MB Anderson, JK Noel and JC Sprott 
  3. Lotka-Volterra Dynamics – An introduction by Steve Baigent 
  4. Publications of Edward Lorenz 
  5. VTK The Visualization Toolkit 
  6. wxWidgets 
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