# Complexity & Nonlinearity

In this video we give a quick overview to complexity theory, systems thinking and complex systems that will form the foundations to our discussion during the rest of the course.

### Nonlinearity

Nonlinearity is at the heart of complexity science and complexity economics, so before starting the course we just want to lay down a quick working definition of what we mean by this term as it will be a pervasive theme throughout the course. Nonlinearity may be defined in different ways depending on the context but in order to give a formal definition to it we will borrow the formalization from physical, where a nonlinear system is defined as one that defies the superposition principles, meaning with nonlinear phenomena the principle of homogeneity and additivity breakdown, I will now describe what this means.

Starting with additivity; the principle of additivity states that when we put two or more components together, the resulting combined system will be nothing more than a simple addition of each componentâ€™s properties in isolation. The additivity principle, as attractively simple as it is, breaks down in nonlinear systems, because the way we put things together and the type of things we put together affect the interactions that make the overall product of the components combination more or less than a simple additive function and thus defies our additivity principle and we call it nonlinear. There are many examples of this such as putting two creatures together, depending which type of creatures we choose, we will get qualitatively different types of interactions between them, that may well make the combination non-additive, bees and flowers create synergistic interactions or lions and deer interacting through relations of predator and prey, both of these represent either super or sub-linear interactions, the whole will be more or less than the sum of its parts in isolation.

Next the principle of homogeneity, that essentially states that the output to the system is always directly proportional to the input, twice as much into the system, twice as much out, four times as much in four times as much out and so on. The direct implication of this homogeneity principle is that things scale in a linear fashion, which clearly fails to account for the effect that the output of the previous state of the system will have on its current or future state. Put simply our linear model does not deal with feedback loops, inputs and outputs simply appear and disappear without any relation between them.

The homogeneity principle may often work as an approximation, but the underlining fact is that as soon as we put our system into the real world, that is to say into an environment where it operates within both space and time, there will inevitably be feedback loops, as the actions it takes effect its environment with those effects in turn feeding back to affect the future state to the system. This means as soon as we start to deal with the real world, things start to get nonlinear and the more interactions we incorporate into our model, thus making them more robust and realistic, the more nonlinear things are likely to become. The superposition principles breakdown and nonlinearity arises whenever we take into account the nature of the interactions within a system, both between constituent components and over time through feedback loops.