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In this video we will discuss the second source of nonlinearity, what are call feedback loops, where the previous output to the system has some effect on its environment and this then in turn feeds back to affect the current or future input to the system making exponential growth and decay possible.
If we were to draw a model of a linear system it would look something like this, there would be an input to the system, some process and an output, as we can see the input and the output to the system are independent from each other. The value that we input to the system now, is not in any way affected by the previous output. There are of cause phenomena where this holds true, such as the flipping of a coin, the value I will get from flipping a coin now will not be dependent in any way on the value I got the last time I flipped it, in mathematics this is called the Markov property.
But the fact is that many things in our world don’t behave like this, meaning that current input variables to the system are dependent on a previous outputs and current outputs will effect future inputs. The state of the weather yesterday will effect the state of the weather today, the amount of money I have in my account today will through interest effect the amount I have tomorrow and so on. This phenomena where the output of a system is “fed back” to become inputs as part of a chain of cause-and-effect that forms a circuit or loop is called a feedback loop.
A feedback loop could be defined as a channel or pathway formed by an ‘effect’ returning to its ’cause,’ and generating either more or less of the same effect. An example of this might be a dialogue between two people, what you say now will effect what the other person will say and that will in turn feedback as the input to what you will say in the future. A full discussion of the dynamics of feedback loops is beyond the scope of this module, our aim here is to merely touch on how they effect the behavior of a system with respect to nonlinearity.
Feedback loops are divided into two qualitatively different types, what are called positive and negative feedback. A negative feedback loop represents a relationship of constraint and balance between two or more variables, when one variable in the system changes in a positive direction the other changes in the opposite, negative direction, thus always working to maintain the original overall combined value to the system.
An example of this might be the feedback loops that regulate the temperature of the human body. Different body organs work to maintain a constant temperature within the body by either conserving or releasing more heat, through sweating and capillary dilation they counter balance the fluctuations in the external environment’s temperature. Another example of negative feedback loop might be between the supply and demand of a product. The more demand there is for a product the more the price may go up which will in turn feedback to reduce the demand.
We can note the direct additive relationship here when one component goes up the other goes down in a somewhat proportional fashion with the end result being a linear system that tends toward equilibrium. We haven’t yet had the chance to discuss the significance of equilibrium but it plays a very important role in linear systems theory. When we have these additive negative feedback loops the net result is a zero sum game, the total gains and loses combined are zero and we can then define this as the system’s equilibrium or “normal” state, with our models then being build on this assumption of there being an equilibrium. This concept of equilibrium holds well for isolated systems and systems in negative feedback loops, but as we will see this assumption about their being an equilibrium breaks down in nonlinear systems and thus we describe them a being “far-from-equilibrium”.