Please note, that I will often put in links to Wikipedia articles. One has to be careful about how accurate Wikipedia pages are, and in general if you are writing a scientific research paper, Wikipedia articles are not good to reference. However, for an educational set of notes like this, where I think that it will add useful content, I will be doing so. Wikipedia articles should be read with a critical hat on, and if you want to dig further, look at the references in the articles themselves.
I am writing these notes using the Wolfram Mathematica programming language. It means that I can very easily include nice plots and animations which should add to the content. You do not need to know how to use this.
The plan of action for this semester (more or less):
All models are wrong, but some are useful George Box (sort of)
1) Exponential growth:
2) The logistic equation:
3) The pendulum:
All of these differential equations are in terms of time, but a differential equation doesn’t have to be. However, in this course, we will be talking about time...and we’ll see why. The key is in the word dynamics. If I give you a differential equation and tell you that it describes some complicated system, like populations of rabbits and wolves, or planets moving around each other, or neurons firing in the brain, can we figure out something about the behaviour of that system without having to go to the hassle of actually solving the equations? Well, it turns out that very often we can. And that, primarily, is what this second half of MAM1043H is about. We are going to be studying so-called dynamical systems. Systems of interacting elements (things) which are described by differential equations. We will see lots along the way, but the types of things that we might be interested in are:
∘ A planet moving around a star ∘ A population of rhinos interacting with a population of poachers ∘ The propagation of light from a lightbulb through the air ∘ The price of oil ∘ The flow of water through a river ∘ The weather ∘ The evolution of life on Earth ∘ The expansion of the universe ∘ The swinging of a pendulum ∘ The firing of a single neuron in your brain ∘ The firing of all of the neurons in your brain ∘ The growth of the internet ∘ The flow of heat through a piece of metal ∘ The spread of fire through a forest ∘ The running of a computer program ∘ The increase in the number of bacteria in a sample ∘ and so many more!
As you can see, these already come from so many disciplines: physics, biology, economics, computer science, meteorology, sociology, neuroscience...
What do all of these things have in common?
They are all about how something changes in time.
They are all about the interaction of something, either with itself, or in some environment. Let’s look at that list again:
∘ A planet moving around a star: The position of the planet is changing with time, the planet is interacting with the star ∘ A population of rhinos interacting with a population of poachers: The number of rhinos and poachers are changing over time. The rhinos and poachers are interacting with each other, but the rhinos also depend on the amount of food around, and many other factors. The number of poachers depends on economic factors, family dynamics, police enforcement. ∘ The propagation of light from a lightbulb through the air: Light is an excitation of the electromagnetic field. The position of photons changes with time. ∘ The price of oil: The price changes over time. It is dependent on the supply and demand. ∘ The flow of water through a river: We can treat the individual water molecules, or we can talk about the velocity of water at any one point and see how that changes over time.
Now try and complete the list below on your own, as much as you can. There are many ways to answer this, and most of them are very complicated systems. It's more a matter of getting you to think about it than getting a fixed answer.
∘ The weather ∘ The evolution of life on Earth ∘ The expansion of the universe ∘ The swinging of a pendulum ∘ The firing of a single neuron in your brain ∘ The firing of all of the neurons in your brain ∘ The growth of the internet ∘ The flow of heat through a piece of metal ∘ The spread of fire through a forest ∘ The running of a computer program ∘ The increase in the number of bacteria in a sample ∘ and so many more!
Now try and write down some dynamical systems of your own...ie. interacting things which change in time.
You may ask the question “What is a system which is not dynamical”. Well, the definition of system, which I think is fair here, is one from Wikipedia (though many dictionary definitions are similar):
“A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole.”