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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The essence is that small changes in the start of a system can cause big variations at the end.
Imagine rolling a marble along the ground. On a firm even surface you can pretty much predict where it will go based on where you roll it from, and if you roll it again from one inch to the left with the same power you can expect it to end up one inch left from where you rolled it last time. But add some bumps, dips, valleys, grease, sand, etc, to the ground, and rolling the marble from a slightly different spot or with slightly different strength means it will end up somewhere you wouldn’t expect based solely on the difference in that initial roll.
Roll some dice.
The result you’ll get it is difficult to determine, even if you know how for the floor is, how fast the dice are spinning and moving, etc.
Chaos Theory is the idea that alter those initial conditions just a little, and you’ll get a very different outcome in a way that is extremely difficult to predict for a something as simple as a dice roll.
Real life, with more initial conditions and more sensitivity to those conditions, is way harder to predict.
The [Wikipedia article on the butterfly effect](https://en.wikipedia.org/wiki/Butterfly_effect) has more information that you probably want to know, comprehensively written in a fairly comprehensible way.
The article even includes a video of six actual chaotic systems demonstrating the effect of small perturbations of initial conditions can result in dramatic variations in ending conditions.
Chaos Theory says that small actions (butterfly flaps wings) can cause large unpredictable consequences (hurricane) when the chain of consequences is very long (imagine a very long line of dominoes falling down).
Here’s a simple real life example. My grandfather met my grandmother in a cinema after a movie. My grandfather only went to the movie because his friend invited him. His friend invited him only because they had become friends after a school fight. The fight started because my grandfather was accidentally hit by an inaccurate spitball and retaliated against the wrong kid. Essentially, the fact that my family exists was caused by someone having a bad spitball aim.
Think about your life. There are tons of examples of how small random events lead to large consequences.
Chaos theory is now called Complexity Theory.
In the simplest terms, CP is about finding underlying order in seemingly complex or random sequences or events. A common example is predicting weather. Another might be predicting how crowds will behave in a riot.
However, CP also applies to the opposite phenomenon; understanding the complexity of otherwise simple systems. Like pool balls moving on a flat pool table. Theoretically, if you know the weight of the balls and the angle and power of the shot, you could predict where the balls will all be a thousand turns in the future. Except you can’t, because of tiny flaws in the round surface of the ball or the flat table, or wind resistance, or friction, all of which turn an otherwise simple system into a complex one. These tiny flaws will have long term effects on the outcome.
So it is about making the simple things complex and the complex simple.
Chaos theory is like when even small actions can have big consequences, kinda like how a tiny butterfly flapping its wings can set off a huge storm far away
It’s pretty wild!
I think you’ve already had it answered, but Jeff Goldblum as Ian Malcolm in Jurassic Park gives a fairly nice, simplistic explanation of it that you could recreate.
As he says, trying to predict how a drop of water is going to roll on the hand is very difficult, if not impossible. There are simply too many factors at play (imperfections in the skin, differences in the size of water drops, variations in blood vessel size as blood flows in and out, etc.). As more factors come into play in any system, outcomes become increasingly difficult to predict.
The essential concept is that Big things can have Small Beginnings.
The idea of the Butterfly-Effect isn’t that the flap of the wings is magnified up into a hurricane-force wind, it’s that there is a chain of events set in motion which leads to something disproportionately bigger, and the longer the period of time you look at, the greater the spread of effects are.
The classic example is that you (a time-traveller) travel back to the age of the dinosaurs, and while you’re there you swat a bug that was going to bite you.
That bug’s descendants through time number countless billions or trillions of insects.
Its descendants will have bitten people and animals throughout millions of years of history.
All of that has stopped, and while doubtless people are still going to get bitten by bugs, it won’t be in the same way, or the same times.
The Spanish explorer bitten by an insect and killed by fever lives on, and becomes a political figure, and now we have recognisably different history with more or fewer wars, and different alliances.
When you come back to your own time-period, it may not be recognisable. The Americas might predominantly be spanish-speaking, or Spain might be gone entirely. Conquered by the Incans.
That’s Chaos Theory.
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.
Let’s leave the chaos theory as a whole for a bit and concentrate on the weather part.
The way I had it explained years ago is this. The equations we currently use for modeling and predicting weather are “numerically unstable”.
What that means is normally when you do calculations, you expect a small change in numbers on one side to affect the result in a small way. For example – if you want to know how far you’ll travel after 10 hours of going 100 km/h you’ll get 1000 km. If you change the speed by 0,1 and have 100,1 km/h, the final answer is 1001 km, so it changes just as much as the input.
In case of numerically unstable equations, a very minor change on one side can cause a difference in levels of magnitude. In the example I have it would mean that for some reason changing the speed just 0,1 meant you traveled 10 000 or 100 000 km.
That of course doesn’t happen with speed and distance, but it does happen with our weather models.
The butterfly and hurricane are just a metaphor to illustrate that. No butterflies actually cause hurricanes, from what I understand it says more about how insufficient our current models are than about some mystical characteristic of the world we live in.
As a side note, it hapens a lot with physics and maths. The way they work are often misconstructed to give people this weird mystical idea which is actually very far from the truth. Main examples would be the number pi or the fibonnaci sequence, there is nothing mystical and spiritual about them, there are just mathematical concepts that we use to describe various, rather mundane things.
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