Dynamical systems are sets of mathematical equations which describe how variables change in time. Most of the interesting dynamical systems have evolving states which depend somehow on the current state. This can be graphed in what’s called a phase portrait, which involves taking the system’s variables on the x and y axis, and plotting curves which show how they evolve away from their initial state.
Here are some examples, which I created as animations in Processing. Each solution is a particle, which is dropped at a random x/y position, which then determine the evolution of the particle's position to the next position, and so on.
You will see that some systems like to pull all the particles towards a single path, whereas some are messy and diverging. These differences are important to understanding why some things in nature are stable and unchanging, whereas some never sit still.
The Predator-Prey Equations
The first dynamical system is the Lotka–Volterra, or ‘predator-prey’ equations. It shows how the number of foxes in a forest (vertical axis) varies with the number of rabbits (horizontal axis) as they all make babies and the rabbits are eaten by the foxes.
As the foxes eat the rabbits, their population grows, but eventually this rate reduces then reverses as the rabbit population falls and the foxes starve. As the foxes starve, fewer foxes means more rabbits, which causes the fox population to bounce back and the rabbit population to decline… and so on, producing the cyclical behaviour you can observe in the animation.
We can see that there’s an equilibrium point about which the numbers of both tend to be stable, and that further from this point, the populations undergo wild oscillations with each nearly being wiped out before growing enormous and repeating the cycle.
The Watt Governor
Despite sounding like the utterances of a cheery Victorian street urchin, the Watt Governor is in fact a simple device for regulating an engine’s speed. It’s purely mechanical, consisting of a motor hooked up to a pulley which controls a fuel supply. If the motor spins fast, the fuel intake goes down, slowing it; if the motor spins slow, the fuel supply gets increased, and the motor speeds up. This establishes a stable equilibrium and regulates the engine’s speed.
As a point of interest, the Watt Governor is the traditional model invoked by dynamical systems theorists and philosophers when proposing dynamical theories of mind.
In the animation, the horizontal axis is the angle of the Watt Governor’s arms - the greater the angle, the more the slowing effect. The vertical axis is the rotational speed of the flywheel. We can see the behaviour is once again cyclical, but this time the equilibrium point is attractive, and all solutions tend towards it. This is the reason for the term ‘Governor’ in ‘Watt Governor’; unlike the predator-prey model, this system likes to settle towards its equilibrium values (which makes sense - a chaotically oscillating engine isn’t very useful).
Limit Cycles
A concept which comes up often, both in Chaos theory and in the modelling of oscillatory systems, is that of a limit cycle. A limit cycle is a looping trajectory which a large number of possible systems, regardless of their initial values, will settle into. The example provided is known as the van der Pol oscillator.
Some fun ones
These phase portraits came up as part of my research tinkering with the equations. They don’t correspond to anything in nature as far as I know, but they make for pretty pictures. I call them “Octopus” and “Vertigo”.
