I remember reading that a butterfly on the otherside of the world can cause a hurricane on the opposite side, and it’s down to chaos theory, could someone explain what chaos theory is please? Thanks
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Chaos theory was originally discovered by Edward Lorenz who was trying to model the weather with an early computer. Now somehow he lost the original data and he used his backup to run the simulation again. But he got a massively different result. As it turns out he had specified the input parameters very accurately but the backup only stored them up to some 6 decial digits.
So for some reason the system he was simulating was so sensitive to initial conditions that evem just missing the last few decimal digits the result was completely different.
And so chaos theory was born. We talk about a system being chaotic when the system is highly sensitive to initial conditions. We usually study these systems by studying their phase space. A phase space is when you plot positions and velocity. For a harmonic oscillator the phase space is circles around the origin.
So we would look at trajectories in this phase space and see where they end up. On common way that chaos can happen is through circles. You have fixed points they are cycles of 1 and you have two cycles where the system cycles between two points and so on for three and four. This cycling will happen when you change some parameter of the system. And the system falls into more and more cycles until you get infinite cycles which is now chaos.
In dissipative systems you often have a chaotic attractor. So initially your trajectory is pretty ordinary like spin a double pendulum around it’ll make nice circles and then as it slows down its motion will be chaotic. In the phase space the trajetory landed on the attractor. Same example but let the double pendulum wind down a bit until it starts to move like an ordinary pendulum. The chaos dissappears, this is called transient chaos.
You can study these attractors or filaments in non-dissipative systmes based on their geometry. They are fractals.
Its all fun and games but still we lack some formal definition of what chaos is. So if you are ready for something a bit more complex:
In a phase space we have hyperbolic fixed points. And around these points if we make a linear approximation we find that (for a 1D system) there are two eigen-directions. In one directions all trajectories move away form the point and in another all trajectories move towards the point. Looking at the bigger picture we find stabil and instabil sets of trajectories. The instabil trajectories always move away from the point while the stabil ones always move towards the point. So what happenes (since these trajectories are on top of each other) when a stabil and an instabil trajectory intersect? The intersection point has to move cloaser and further away from the point. It has no option but to get to another point like this. There are an infinite number of these intersection and so the motion of these points is some random walking between them. This is chaos.
If you are interested in the deeper ideas of chaos I recommend reading into the Duffing oscillator. Its the harmonic oscillator of chaotic systems, you can actually calculate stuff and show things.
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