Global warming. It's not worth the risk

The Butterfly effect has a technical name. It is “sensitive dependence on initial conditions.” This does not mean the system is chaotic in the usual sense of the word, it means that the system cannot be predicted because we would have to do every calculation with perfect information continuously. Each calculation depends on the calculation before so there are no short cuts. Even the slightest variation will change the course of the system.

This means that every little we do will have an effect on the future, and there is no way of predicting how.

Dreams about controlling the weather have long since gone. The seven-day forecast on the television is at best a wild guess. But there is a glimmer of hope in all this chaos. The hope lies within the form of the Lorenz attractor.

The Lorenz attractor shown plots the 3D position of point over time for a stream of chaotic data. A 2D representation is shown above, and it shows also why butterflies are so popular in chaos theory.

It is total chaos. Nothing is predictable and nothing ever repeats, but is does stay within certain limits and seems to move according to an attractor (or two) of some sort. If you go onto the internet you will find any number of Lorenz attractor generators. Some you can change the input and see how the chaos changes. I have one as a screen saver. I watch it for hours when I should be working. It produces very beautiful patterns.

An attractor in the sense used is a point that the data tends towards. The Data in a non-chaotic system would tend towards a point and then stop there. A pendulum will swing backwards and forwards in decreasing arcs until it stops in the vertical position, attracted by gravity. The length and timing of each arch is predictable down to the point where motion ceases.

But in the chaotic system nothing is predictable except by inspection it does seem to be kept within certain limits by unseen attractors.

One way of looking at attractors is a system’s natural state. In a non-chaotic system the system will reach and stabilise at the attractor in a predictable manner. In a chaotic system the system will never reach a closed state at the attractor and will never be predictable.

We cannot predict the weather seven days in advance, but we do have some idea that it should stay within certain limits over a longer time span even if never repeats exactly. It will stay within these limits unless the attractor is changed by an overall change in the conditions of the system. We have the conditions for spring, summer, autumn and winter within general limits unless we do something to change the overall conditions, like pumping billions of tons of carbon dioxide into the atmosphere.

If we keep pumping carbon dioxide into the atmosphere the weather could flip to another set of rules. That is the transitive nature of our chaotic weather pattern.

Transitive systems

A transitive system is a system that can work with different sets of rule, but not at the same time. An example of a chaotic transitive system is the human heart. Even though there is a pattern no heartbeat is exactly the same as the last. There is always a slight variation based in part on the heartbeat that went before. This is a perfect example of chaos not being chaotic.