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.

Later edit: Since I mentioned climatology and media cherry picking, I think I should also add some information I directly checked on the subject. Not so long ago, a lot of propaganda was circulated, about Earth going Venus or something like that. I’m subscribed to all sorts of groups that are supposed to present scientific news, but since it’s media, they cannot help themselves to add their bias on top. So, exaggerated titles as usual, you should get the picture. Despite the fact that the authors clearly stated that Earth is not going Venus, the titles ‘suggested’ otherwise. The comments were precious, as always. I was annoyed enough by the pseudo scientific propaganda to look into the actual computer model. It was a simple model (too simple to be able to pretend to simulate the real world), easy to check out because they put the code on GitHub. I’ve looked five minutes into it, until I found this: Climate computer model bug. It’s a serious bug, to compute optical thickness with H2O instead of CO2 after CO2 was so badly presented as the devil is not something nice. Errare humanum est, but in this case it is diabolical. Not intended, that is almost certain, but nasty nevertheless. Why can that happen? Well, because it’s a pseudo science, not a science. A real science would have means to test the model against the real world and find that it does not fit the model. A pseudo science that does not have that ‘luxury’ would let such errors pass as the absolute truth. There is an advantage of not having to put your model to the test. Or maybe not.


  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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