Chaotic Systems

Systems Thinking, Week 10

After spending last week talking about the complexity of systems on the border between stability and chaos, this week we’re diving into chaotic systems. First, let’s be really clear about definitions: while “chaotic” in everyday language mostly means unpredictable and unstable, but scientists mean something more precise. I’ll be honest, the exact definition is pretty confusing to me, because I hear some conflicting ideas at the very foundation: Crutchfield and friends talked about it in 1986 as randomness generated by (potentially) simple deterministic systems. On the other hand, watch the beginning of this Nova special on the topic, and they talk about the Chaos Game, where simple rules plus the rolling of a die¹ lead to a consistent, ordered pattern, the Sierpinski Triangle. This makes me think that the concept includes two basic interconnected relationships: simple rules → seemingly random behavior, and (seemingly or actually) random behavior + simple rules → surprising patterns and order. Note that the definition of “surprising patterns” has to exclude the emergence of simple order from randomness; this is the purview of statistics, with results like the Central Limit Theorem.

One of the properties of chaotic systems is that they are sensitive to initial conditions; that is, tiny changes at the beginning will lead to massive changes over time in how things play out. As Edward Lorenz, one of the pioneers of the field, puts it: the present determines the future but the approximate present does not approximately determine the future. This makes chaotic systems impossible to predict, because any measurement involves some degree of error. For the record, the system is sensitive at every point, not just the beginning², so any perturbations or intervention along the way will also change the future in unpredictable ways.

Another property of chaotic systems is transitivity. Not only do small changes explode over time into huge differences in outcomes, but there’s a boundary around the system, and, with enough time, conditions in any one part of that boundary will end up in any other part. Basically, the system will eventually end up visiting every possible configuration. In the classic double pendulum example, the system describing the position of the outmost edge at a given time is bounded by the lengths of the arms and the total energy of the pendulum (potential and kinetic), in a shape described by an outer arc (when the arms are both extended in the same direction) and an inner arc (when the second arm is fully turned in). A pendulum in motion in any region within that boundary will eventually reach any other region you choose inside that boundary.

Because it can create apparent randomness from simple deterministic systems, chaos is a useful way that living systems can create stochastic noise. For example, the Nova special presents (around 44:25) the case of muscles: balance and stability is the result of lots of tiny muscles constantly counteracting one another, so if the micromovements are chaotic, there’s no risk of developing the amplifying periodic behavior of a tremor. Because of transitivity, chaos can be great to add to search strategies: no matter where you start, given enough time you’ll explore the entire space.

One of the emergent properties of chaotic systems is self-similarity: chaotic processes tend to create trails/impacts/artifacts that where each part looks like the whole. This shows up in all sorts of natural settings, including the shape of mountains and clouds. These self-similar shapes, including the Sierpinski Triangle mentioned at the beginning, are called fractals and have plenty of neat mathematical properties; I’ll just say that they are a perfect example of the kind of emergent behavior that flows from the rules of the system.

We already have a good word that in both its everyday and technical meaning conveys the behavior of chaotic systems: turbulence. Consider the example of wildfires. Normally, a wild ecosystem acts as a complex system, with lots of activities that create change and balance. Within this system, things like lightning strikes usually cause local damage to trees etc. However, once the system moves beyond its limits (specifically moisture and heat), the same lightning strike can start a fire that spreads (extreme non-linearity). Weather, the original chaotic system, influences how fast and in what direction it spreads, and prediction of its behavior is extremely difficult more than a few hours ahead (leading to graduated warnings covering big geographic areas). People actively attempt to control the fires, but unexpected things happen all the time. The chaos can last as long as there’s fuel.

Obviously this is just scratching the surface of what’s an entire field of mathematics, but it shows the relevance of chaos applied to systems thinking: it doesn’t take much complexity to create turbulence, no amount of measurement will help you make long-term projections, but it may be possible to derive some of the underlying emergent structure and bound the chaos.

1. I’m aware that you can think of the roll of a fair die as an example of the first kind of chaos, where the simple (deterministic) laws of physics acting on a chance cube create what seems to be a uniform discrete distribution. However, the example works even if you used a hypothetical pure random number generator.

2. As an example of initial sensitivity only, imagine dropping a marble at the peak of a mountain: two marbles dropped just inches apart could end up miles away. However, the difference shrinks dramatically when you move the starting point just a few feet down one of the slopes.