Exponentials & Power Laws

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Exponentials are a signature key of nonlinear systems, unlike linear growth exponential grow represents a phenomenon where the actual rate of growth is growing itself to generate an asynchronous development with respect to time and some very counter-intuitive events. In this module, we will discuss the dynamics of exponentials and their counterparts power laws that represent a power relation between two entities.

Part of our definition for linear systems is that the relationship between input and output is related in a linear fashion, the ratio of input to output might be 2 times as much, 10 times as much or even a thousand times it is not important this is merely a quantitative difference, what is important is that this ratio between input and output is itself not increasing or decreasing. But as we have seen when we have feedback loops the previous state to the system feeds back to effect the current state thus enabling the current ration between input and output to be greater or less than its ratio previously and this is qualitatively different.

This phenomenon is captured within mathematical notation as an exponential. The exponential symbol describe how we take a variable and we times it by another not just once but we in fact iterate on this process, meaning we take that output and feed it back into compute the next output, thus the amount we are increasing by each time itself increases. So lets take a quick example of exponential grow to give you an intuition for it.

Say I wish to create a test tube of penicillin bacteria. Knowing that the bacteria will double every second, I start out in the morning with only two bacteria hoping to have my tube full by noon. As we know the bacteria will grow exponentially as the population at any time will feed into effect the population at the next second, like a snowball rolling down a hill. It will take a number of hours before our tube is just 3% full but within the next five seconds as it approaches noon it will increase to 100% percent of the tube.

This type of disproportional change with respect to time, is very much counter to our intuition where we often create a vision of the future as a linear progression of the present and past, we will be discussing farther the implication of this type of grow later when we get into the dynamics of nonlinear systems but for the moment the important thing to note here is that in exponential growth the rate of growth itself is growing and this only happens in nonlinear systems, where they can both grow and decay at an exponential rate.

Exponentials are also called powers and the term power law describes a functional relationship between two quantities, where one quantity varies as a power of another. There are lots of example of the power law in action but maybe the simples is the scaling relationship of an object like a cube, a cube with sides of length a will have a volume of a3 and thus the actual number that we are multiplying the volume by grows each time, this would not be the case if there was a simple linear scaling such as the volume being 2 times the side length.

Another example from biology is the nonlinear scaling within the metabolic rate vs. size of mammals, the metabolic rate is basically how much energy one needs per day to stay alive and it scales relative to the animals mass in a sub-linear fashion, if you double the size of the organism then you actually only need 75 % more energy. One last example of the power law will help to illustrate how it is the relationships between components within a system that is a key source of this nonlinearity

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