Non-Linearity

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.