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Non-linear systems are ones where the scale or size of effects is not correlated with the scale of causes, making  them very difficult to predict.

Non-linear systems are ones in which a small change to initial conditions can result in a large scale change to the system's behavior over the course of time. This is due to the fact that such systems are subject to cascading feedback loops, that amplify slight changes. The notion has been popularized in the concept of 'the butterfly effect'.

The behavior of non-linear systems is governed by what is known as {{positive-feedback}}. What is interesting about positive feedback and the dynamics of non-linear systems is that they disrupt our normal understanding of causality. We tend to think that big effects are the result of big causes. Non-linear systems do not work that way, and instead a very small shift in initial conditions can result in massive system change. It therefore becomes very difficult to determine how an input or change will affect the system, with small actions inadvertently leading to big, unforeseen consequences.


Clarifying Terminology: Positive feedback does not imply a value judgement, with 'positive' being equated with 'good'! Urban decay is an example of a situation where positive feedback may lead to negative outcomes. A cycle of feedback might involve people divesting in a neighborhood, such that the quality of the housing stock goes down, leading to dropping property values at neighboring sites, further dis-incentiving improvements, leading to further disinvestment, etc.

History Matters!

The non-linearity of complex systems make them very difficult to predict, and instead we may think of complex adaptive systems as needing to unfold. Hence, {{history-matters}}, since slight variances in a system's history can lead to very different system behaviors.

Example:

A good example of this is comparing the nature of a regular pendulum to a double pendulum.

In the case of a regular pendulum,  regardless of how we start the pendulum swinging, it will stabilize into a regular oscillating pattern. The history of how, precisely, the pendulum starts off swinging does not really affect the ultimate system behavior. It will stabilize in a regular pattern regardless of the starting point, that can be replicated over multiple trials. The situation changes dramatically when we move to a double pendulum (a pendulum attached to another pendulum with a hinge point). When we start the pendulum moving the system will display erratic swinging behaviors - looping over itself and spinning in unpredictable sequences. If we were to restart the pendulum swinging one hundred times, we would see one hundred different patterns of behavior, with no particular sequence repeating itself. Hence, we cannot predict the pendulum's behavior, we can only watch the swinging system unfold. At best, we might observe that the system has certain tendencies, but we cannot outline the exact trajectory of the system's behavior without observing it:

watch the double pendulum!

We can think of the difference between this non-linear behavior and linear systems: if we wish to know the behavior of a billiard ball being shot into a corner pocket, we can calculate the angle and speed of the shot, and reliably determine the trajectory of the ball. A slight change in the angle of the shot leads to only a slight change in the ball's trajectory. if the behavior of a billiard ball on a pool table were like that of a complex system. Accordingly, people are able to master the game of pool based on practicing their shots! If pool behaved like a complex system it would be impossible to master: even with only the most minute variation in our initial shot trajectory, the ball would find its way to completely different positions on the table with every shot.

System Tendencies

That said, a non-linear system might still exhibit certain tendencies. If we allow a complex system to unfold many times (say in a computer simulation), while each simulation yields a different outcome (and some yield highly divergent outcomes), the system may have a tendency to gravitate towards particular regimes. Such regimes of behavior are known as Attractor States. Returning to the pendulum, in our single pendulum experiment the system always goes to the same attractor, oscillating back and forth. But a complex systems features multiple attractors, and the 'decision' of what attractor the system tends towards varies according to the initial conditions.

Complex systems can be very difficult to understand due to this non-linearity. We cannot know if a 'big effect' is due to an inherent 'big cause' or if it is something that simply plays out due to reinforcing feedback loops. Such loops amplify small behaviors in ways that can be misleading.

Example:

If a particular scholar is cited frequently, does this necessarily mean that their work has more intrinsic value then that of another scholar with far fewer citations?

Intuitively we would expect that a high level of citations is co-related with a high quality of research output, but some studies have suggested that scholarly impact might also be attributed to the dynamics of positive feedback: a scholar who is randomly cited slightly more often than another scholar of equal merit will have a tendency to attract more attention, which then attracts more citations, which attracts more attention, etc.. Had the scholarly system unfolded in a slightly different manner (with the other scholar initially receiving a few additional citations), the dynamics of the system could have led to a completely divergent outcome.  Thus, when we say that complex systems are "Sensitive to Initial Conditions"  this is effectively another way of speaking about the non-linearity of the system, and how slight, seemingly innocuous variation in the history of the system can have a dramatic impact on how things ultimately unfold.

Some Urban Thoughts

See also far from equilibrium.

 

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In Depth... Non-Linearity

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Nonlinear Dynamics & Chaos

Follow along with the course eBook: https://systemsinnovation.io/books/ Take the full course: https://systemsinnovation.io/courses/ Twitter: http://bit.ly/2JuNmXX LinkedIn: http://bit.ly/2YCP2U6 For many centuries the idea prevailed that if a system was governed by simple rules that were deterministic then with sufficient information and computation power we would be able to fully describe and predict its future trajectory, the revolution of chaos theory in the latter half of the 20th century put an end to this assumption showing how simple rules could, in fact, lead to complex behavior.

This is a list of People that Non-Linearity is related to.

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