Having now laid down our foundations this is where our discussion on nonlinearity really starts. We will talk about why and how linear systems theory breaks down as soon as we have some set of relations within a system that are non-additive, which appears to be often the case in the real world, we also look at how feedback loops over time work to defy the homogeneity principle with the net result being nonlinear behavior.
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</p><p>Transcription excerpt:
</p><p>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 value to start our discussion by asking 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 defined a linear relation.
For centuries science and mathematics has been focused upon these simple linear interactions and orderly geometric forms, that can be describe 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 scientist 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 has lumped them with being defined by what they are not.
In the previous section we discussed the key characteristics of linear systems, what is called the superposition principals, we can then defined nonlinear systems as those the defy the superposition principals, meaning with nonlinear phenomena the principals of homogeneity and additively break down, but lets take a closer look at why this is so.
Starting with additively, as we have already discussed additively 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 additively principal, 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 effect the interactions that make the overall product of the components combination more or less than a simple additive function and thus defies our additively principal and we call it nonlinear.
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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 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.
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