A nonlinear system is a type of system that defies the superposition principles of homogeneity and additivity. Feedback loops between components within nonlinear systems and over time give rise to synergies or interference that make the output to the system less or more than the sum of its parts and thus nonadditive.1
Although it is often said that nonlinear systems describe the vast majority of phenomena in our world. They have unfortunately been designated as alternatives, being defined by what they are not. It might be of interest to ask why is this so? The real world we live in is inherently complex and nonlinear, but from a scientific perspective, all we have is our models to try and understand it. These models have inevitably started simple and developed to become more complex and sophisticated representations. When we say simple, in this case, we mean things that are the product of direct cause and effect interactions. With these simple interactions, we can draw a direct line between cause and effect and thus define a linear relation. For centuries science and mathematics have been focused upon these simple linear interactions and orderly geometric forms that can be described in beautifully compact equations. Not so much because this is how the world is, but more because they are by far the easiest phenomena for us to encode in our language of mathematics and science. It is only in the past few decades that scientists have begun to approach the world of systems that are not linear. Thus, their late arrival on the scene and our lack of understanding of what they really are have lumped them with being defined by what they are not.
Linear systems are characterized by what is called the superposition principles. We can then define nonlinear systems as those the defy the superposition principles, meaning with nonlinear phenomena the principles of homogeneity and additivity break down. 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 component’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 the additivity principle and we call it nonlinear. There are many examples of this such as putting two creatures together. Depending on which type of creatures we choose, we will get qualitatively different types of interaction 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.
Of course, this is all very intuitive to us. Learning about these nonlinear interactions between things is all part of growing up and learning to have a normal common sense, but the problem is actually formulating this in the language of science and mathematics. Whereas we can easily and rigorously study the properties of elements in isolation by taking them into a laboratory or some other isolated environment, it is more difficult though for us to know why, or when, or if, elements will have some special interaction. And not only this, but these interactions often create novel and surprising new phenomena through the process of emergence. Imagine you have to play the role of a matchmaker between two people you know. You may know very well what they are like separately but it would be a lot more difficult for you to tell if they would hit it off when you introduce them to each other. To take another example, who could have imagined that when we put hydrogen and oxygen together we would get water? And not only that but because of the weak hydrogen bonds between them that this new substance would, in fact, have the property of wetness. These examples should illustrate how nonlinearity arises from the non-additive nature to the interactions between things when we combine them.
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 underlying 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 affect 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 models – thus making them more robust and realistic – the more nonlinear things are likely to become.
The immediate analogy that springs to mind of this is the so-called “limits to growth model.” Within the industrial age paradigm that was heavily influenced by linear systems thinking, there was or still is the belief in continuous progress without regard to the effect that the current actions of the system will have on its natural environment, and how these will feed back to affect the future input variables to the system. But as soon as we begin to conceive of this economic system within its environment, we quickly come to the conclusion that this infinite scaling is not possible because the environmental effect will inevitably feedback to constrain the future state of the system at a certain “limit to growth”. Thus, we can see how the homogeneity principle breaks down and we get nonlinear behavior. The superposition principles break down and nonlinearity arises whenever feedback interacts within a system – and feedback loops over time – are present.