Reboot No.3

In September 2013 I started putting together this website because I wanted a place to “gather my thoughts” on Chaos Theory.

Back then I explained Chaos Theory as being about the “Coarse Damping to Equilibrium” meaning that chaos describes the type of behaviour that emerges when a system has difficulty fine-tuning its way to an orderly equilibrium (thanks to the aggressiveness of their internal dynamics)…

This explanation didn’t seem to gain much traction with the world at large, so in January 2016 I tried a different approach. 

This time I decided to explain chaotic behaviour as “Incompressible Dynamics”: meaning that chaotic behaviour is behaviour that is not easily mathematically compressed or verbally summarized…

To my mind this piece of phraseology captured chaos very neatly; because it relates chaotic behaviour to how difficult it is to explain that behaviour.  Chaotic behaviour is effectively difficult to explain behaviour.

The easiest behaviour to explain is a complete lack of behaviour – the thing or system, or person is doing nothing.  The next easiest is constant behaviour, like constant motion, or a constant oscillation, or constant periodic behaviour.  Such behaviour is easy to describe — night always follow day, day always follows night, etc.  However as things become a little more complicated they become a little harder to describe, both mathematically and verbally.

Physics is all about explaining behaviour, ideally in mathematical form.  The trouble is that the more complicated the behaviour, the harder it is to put some mathematics on it.  So much so that for complicated systems scientists usually resort to using averages.  Averages allow us to forget about the details and concentrate on the system as a whole…

Chaos Theory is really all about the behaviour of systems where averages don’t seem to work.  Chaos Theory deals with all types of system behaviour, both orderly and chaotic; but the term “chaos” within these systems refers to behaviour that doesn’t seem to exhibit any form of consistency (i.e. this behaviour doesn’t settle down to some form of average behaviour – and consequently we cannot compress these dynamics into some form of simple mathematics).  Chaos is effectively describing systems with constantly changing dynamics…

This approach seems to be better; got a little more traction, but still didn’t really catch fire…

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Anyway over the last 3 years or so I have been distracted with other things, but during this time I have also come to the conclusion that what Chaos Theory is ultimately about can essentially be boiled down to 3 key ideas.

  1. Chaos theory is about the Mathematics of Dynamic Equilibrium – in other words, the concept of mathematical equilibrium is not limited to pure stability, but can come in many different forms ranging from stable to dynamic…
  2. Chaos Theory is about Emergent System Behaviour – and the type of system behaviour that ultimately emerges is determined by the strength of system’s internal dynamics
  3. Chaos Theory is about the Existence of Mathematical Attractors (both orderly and chaotic) – which give the impression that there is an invisible hand at work in pulling the system to a particular form of equilibrium…

This new perspective probably demands another reboot of this website, which will require that I review (/alter/correct) a number of the fixed pages on this website.  Unfortunately I am a bit short on time at the moment so I will have to get to that in due course.

In the meantime, I have some more pressing work to attend to…

Author: Kieran D. Kelly

Experimental and Applied Mathematician, Specialist in Nonlinear Systems and Dynamics…