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A Misunderstood Complexity?
Topic: Chaos
by Tim, 2019 Cohort
Taken at face value, the concept of ‘Chaos’ appears a pervasive, yet elusive idea. It is entangled in everything, yet tricky to define specifically. For example, a common explanation of Chaos is that it is merely the opposite of Order. If Order is that which exhibits logical sequence and structure, then Chaos, conversely, must lacks these qualities. However, this is an inaccurate characterisation. In truth, that which is chaotic may simultaneously showcase patterns, feedback loops, and a relation to its conditions of creation. The complexity of the human brain, whilst intricate, is nonetheless a tangible tapestry of cause and effect. The apparent randomness of Chaos is, when scrutinised, recognisable.
What generates the label of ‘Chaos’ then is not the absence of order. Rather, it is the complexity that occurs as an inevitable by-product of approximation.
That is: ?Chaos exists when the present determines the future, but the approximate present does not approximately determine the future.”
A Case Study#
In understanding this definition, it is worth considering the story of Edward Lorenz, a prominent meteorologist during the 20th Century. One day, whilst running his usual weather simulations, Lorenz took a shortcut. In his simulation computer’s memory, six decimal places were stored, 0.506127. To save printing space, Lorenz rounded these decimals to 0.506, a difference of one part in a thousand. On paper, it was an insignificant change. In practice, it was profound:
As can be seen, the rounded and non-rounded predictions at first aligned closely in Lorenz’s model. However, as one line began to lag, this tiny error repeatedly multiplied itself. Ultimately, the results became so divergent as to put the two different forecasts in distinctly separate realities.
Implication#
When it comes to analysing complex systems, it is generally accepted that a chain of events will have pressure points that can magnify small changes. However, Lorenz’s results showed that such points exist everywhere. Given enough scale, any approximation error will produce unique irregularity and unpredictability. Every point is a potential turning point.
Given this insight into Chaos conceptually, this understanding can be represented with a set of more formalised principles.
- Sensitivity to Initial Conditions- The notion that ?each point in a chaotic system is arbitrarily approximated by other points with significantly different future paths. Thus, an arbitrarily small change may lead to significantly different futures”, a phenomenon also known as the ‘Butterfly
- Topological Mixing- The notion that a system evolves over time so that ?Any given region of phase space eventually overlaps with any other given region”. A good example for visualising this is the mixing of coloured dyes together, observing how each dye melds with one another when
- Density of Periodic Orbits- The notion that every point in a systems space is approached closely by periodic orbits. In short, there is a tangible, if unpredictable, boundary of
Whilst these principles are not the exclusive characteristics of Chaos, they are by far the most frequent and, overall, important.
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Disclaimer#
This content has been contributed by a student as part of a learning activity.
If there are inaccuracies, or opportunities for significant improvement on this topic, feedback is welcome on how to improve the resource.
You can improve articles on this topic as a student in "Unravelling Complexity", or by including the amendments in an email to: Chris.Browne@anu.edu.au
Taken at face value, the concept of ‘Chaos’ appears a pervasive, yet elusive idea. It is entangled in everything, yet tricky to define specifically. For example, a common explanation of Chaos is that it is merely the opposite of Order. If Order is that which exhibits logical sequence and structure, then Chaos, conversely, must lacks these qualities. However, this is an inaccurate characterisation. In truth, that which is chaotic may simultaneously showcase patterns, feedback loops, and a relation to its conditions of creation. The complexity of the human brain, whilst intricate, is nonetheless a tangible tapestry of cause and effect. The apparent randomness of Chaos is, when scrutinised, recognisable.
What generates the label of ‘Chaos’ then is not the absence of order. Rather, it is the complexity that occurs as an inevitable by-product of approximation.
That is: ?Chaos exists when the present determines the future, but the approximate present does not approximately determine the future.”
A Case Study#
In understanding this definition, it is worth considering the story of Edward Lorenz, a prominent meteorologist during the 20th Century. One day, whilst running his usual weather simulations, Lorenz took a shortcut. In his simulation computer’s memory, six decimal places were stored, 0.506127. To save printing space, Lorenz rounded these decimals to 0.506, a difference of one part in a thousand. On paper, it was an insignificant change. In practice, it was profound:
As can be seen, the rounded and non-rounded predictions at first aligned closely in Lorenz’s model. However, as one line began to lag, this tiny error repeatedly multiplied itself. Ultimately, the results became so divergent as to put the two different forecasts in distinctly separate realities.
Implication#
When it comes to analysing complex systems, it is generally accepted that a chain of events will have pressure points that can magnify small changes. However, Lorenz’s results showed that such points exist everywhere. Given enough scale, any approximation error will produce unique irregularity and unpredictability. Every point is a potential turning point.
Given this insight into Chaos conceptually, this understanding can be represented with a set of more formalised principles.
- Sensitivity to Initial Conditions- The notion that ?each point in a chaotic system is arbitrarily approximated by other points with significantly different future paths. Thus, an arbitrarily small change may lead to significantly different futures”, a phenomenon also known as the ‘Butterfly
- Topological Mixing- The notion that a system evolves over time so that ?Any given region of phase space eventually overlaps with any other given region”. A good example for visualising this is the mixing of coloured dyes together, observing how each dye melds with one another when
- Density of Periodic Orbits- The notion that every point in a systems space is approached closely by periodic orbits. In short, there is a tangible, if unpredictable, boundary of
Whilst these principles are not the exclusive characteristics of Chaos, they are by far the most frequent and, overall, important.