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Non-Linearity

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'. This effect - the idea that the beating of a butterfly's wings in Brazil, might set off a Tornado in Texas - is counterintuitive because of the scale difference. 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.


This is because 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 are counterintuitive: we tend to think that big effects need to have been created due to 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-incentivizing 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. The pendulum will stabilize in a regular pattern regardless of the starting point, a behavior 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 allowing it to 'play out' in time: 

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.  Accordingly, people are able to master the game of pool based on practicing their shots! If the behavior of a billiard ball on a pool table were like that of a complex system, it would be impossible to master: with even the most minute variation in our initial shot trajectory, the balls would find their ways 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?

Where is this all going?!

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 a different scholar initially receiving a few additional citations), the dynamics of the system could have led to a completely divergent outcome - citation networks may be  subject to historical Contingency, that could have played out differently, with different scholars assuming primary positions in the citation hierarchy. 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. 

Another way of thinking about this is to describe a system's Path Dependency : this is a key concept linked to the idea of non-linearity, that indicates that we need to follow a sequence of the system's unfolding to see what is going to happen. Tied to the idea of a system's path that needs to be followed, is the idea of a Tipping Points, a kind of 'point of no return' where  a system veers from one trajectory to another, thereby closing off other potential pathways. A tipping point can be a system poised at a juncture between two states (either of which could viably unfold - ie VHS or BETA), or a tipping point can be a moment where the pressure on the system is such that it can no longer continue to operate in a particular mode that, until that point, was viable. At that juncture the system needs to move into a different kind of behavioral regime. Water turning to Ice or to steam is a tipping point of this latter kind, where the water molecules move beyond a certain threshold of agitation, and can no longer maintain the state of either solid, liquid, or vapor, beyond that threshold. 


Implications

In many domains of complexity, computer models are the primary tool used to understand these systems. Computers are very effective at emulating the step by step, rule based processes undertaken by multiple agents in parallel, that can result in emergent, unexpected outcomes. There are reasons why this can be very helpful, particularly if the system being modeled can be shown to have a tendency to move towards particular regimes, despite their non-linear features (these system tendencies can be thought of as 'attractors' for the system).

That said, many complex systems do not have specific attractors, or have attractors that change in unexpected ways depending on the environmental context at play. Real world complex systems will gravitate towards 'fit' behaviors, but fitness changes with context, their can be multiple, divergent fit 'solutions',  and the variables governing a system's unfolding can change.

Because of the non-linear nature of complex systems, predictive models, in principle,  are not going to be an effective means to gain insight into ultimate system trajectories. This is not to say that we can't learn from the dynamics that unfold in simulations, only that it is hard to consider them as predictive tools given the inherent uncertainty of these systems.

So what do we do? One answer is that we accept our lack of ability to predict specific outcomes, and try something else. This 'something else' has to do with learning from complexity dynamics so as to gain the tools to enact complexity dynamics:

Enacting vs Predicting.

What is we could set up systems that hold the ability to unfold in ways that lead towards fit behaviors? Rather than build a complex system in a model, what if we could make real things in the world modeled on complexity dynamics? We would have to accept a kind of uncertainty - we won't know where the systems will ultimately look like, but we might still be able to know how the systems will behave. And if we design these systems correctly, they will behave in ways that ensure that energy or resources fueling the system is processed effectively, and that individual agents, are gradually steered into regimes of behavior that maximizes the fitness of all agents, as a whole.

While the precise form such systems take will be subject to contingent, non-linear dynamics, they performance of the system will be something that we can instead rely upon to serve a given purpose.



 

 


Photo Credit and Caption: Image Credit: stephen-hocking-aX9KlIQlrVE-unsplash

Cite this page:

Wohl, S. (2022, 8 June). Non-Linearity. Retrieved from https://kapalicarsi.wittmeyer.io/field/non-linearity

Non-Linearity was updated June 8th, 2022.

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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.

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Logistic Map

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Catastrophe Theory

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Butterfly Effect

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Manifolds

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

Related to the idea of Iterations that accumulate over time

More to come! Learn more →

Known as the butterfly effect - small variations yield large impacts

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Phase space is an abstract concept that refers to all possible behaviors available to an agent within a complex system.

Related to {{degrees-of-freedom}}. Learn more →

A notion in Landscape Urbanism that relates to the notion of an environment's potentiality to be activated in different ways;

can be thought of as connecting to phase space in physics, or the space of possibilities Learn more →

The idea that systems can have more than one stable state.

Early versions of systems theory assumed that systems could be 'optimized' to a single condition. CAS analysis assumes that more than one system state can satisfy optimizing criteria, and so the system is able to gravitate to multiple equilibria.

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Complex system behaviors are highly contingent.

Details of the specific historical trajectory a complex system follows can have a huge impact on system behavior Learn more →

The idea that many possible states or historical trajectories could have equally unfolded

Beyond its day-to-day usage, this term used in now employed in the social sciences to highlight the Path Dependency exhibited in many social systems. This is seen to contrast with prior conceptions like "the march of history", which imply a clear causal structure. By speaking about the work as something contingent, it also begs the question of what other "worlds" might have just as equally manifested, had things been slightly different.

Similar ideas are captured in the ideas of Non-Linearity, {{sensitivity-to-initial-conditions}}, History Matters.

Pictured below: the contingent trajectory of the double pendulum:

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Complex systems do not follow linear, predictable chains of cause and effect. Instead, system trajectories can diverge into wildly different regimes. The moment when a complex system move from one trajectory to another is known as a system bifurcation.

This feature of complex systems means that the behavior of a system cannot be known in advance, but instead needs to be enacted in time. Learn more →

Caramelization half and half robust kopi-luwak, {{fitness}} brewed, foam affogato grounds extraction plunger pot, bar single shot froth eu shop latte et, chicory, steamed seasonal grounds dark organic. see also {{non-linear}}

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This is a collection of books, websites, and videos related to Non-Linearity

A great introduction from the folks at Systems Innovation

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Cellular Automata & Agent-Based Models offer city simulations whose behaviors we learn from. What are the strengths & weaknesses of this mode of engaging urban complexity?

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If geography is not composed of places, but rather places are the result of relations, then how can an understanding of complex flows and network dynamics help us unravel the nature of place?

Relational Geographers examine how particular places are constituted by forces and flows that operate at a distance. They recognize that flows of energy, people, resources and materials are what activate place, and focus their attention upon understanding the nature of these flows.

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Landscape Urbanists are interested in adaptation, processes, and flows: with their work often drawing from the lexicon of complexity sciences.

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Across the globe we find spatial clusters of similar economic activity. How does complexity help us understand the path-dependent emergence of these economic clusters?

Evolutionary Economic Geography (EEG) tries to understand how economic agglomerations or clusters emerge from the bottom-up. This branch of economics draws significantly from principles of complexity and emergence, seeing the rise of particular regions as path-dependent, and looking to understand the forces that drive change for firms - seen as the agents evolving within an economic environment.

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Might the world we live in be made up of contingent, emergent 'assemblages'? If so, how might complexity theory help us understand such assemblages?

Assemblage geographers consider space in ways similar to relational geographers. However, they focus more on the temporary and contingent ways in which forces and flows come together to form stable entities. Thus, they are less attuned to the mechanics of how specific relations coalesce, and more to the contingent and agentic aspects of the assemblages that manifest.

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

A tipping point (often referred to as a 'critical point') is a threshold within a system where the system shifts from manifesting one set of qualities to another.

Complex systems do not follow linear, predictable chains of cause and effect. Instead, system trajectories can diverge wildly into entirely different regimes.

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'Path-dependent' systems are ones where the system's history matters - the present state is contingent upon random factors that governed system unfolding, and that could have easily resulted in other viable trajectories.

Complex systems can follow many potential trajectories: the actualization of any given trajectory can be dependent on small variables, or "changes to initial conditions" that are actually pretty trivial. Accordingly, if we truly wish to understand system dynamics, we need to pay attention to all system pathways (or the system's phase space) rather than the pathway that happened to unfold.

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Left to themselves, systems tend towards regimes that become increasingly homogenous or neutral: complex systems differ - channeling continuous energy flows, gaining structure, and thereby operating far from equilibrium.

The Second Law of Thermodynamics is typically at play in most systems - shattered glasses don't reconstitute themselves and pencils don't stay balanced on their tips. But Complex Systems exhibit some pretty strange behaviors that violate these norms...

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There would be some thought experiments here.

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Non-Linearity
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