Chaos: The Science of the Butterfly Effect

Part of this video is sponsored by LastPass. More on the last pass at the end of the show. The

butterfly

effect

is the idea that tiny causes, like the stripping of a

butterfly

's wings in Brazil, can have huge

effect

s, like causing a tornado in Texas. Now, that idea comes directly from the title of a scientific article published almost 50 years ago. years ago and perhaps more than any other recent scientific concept, it has captured the public imagination. I mean, on IMDB there are not one but 61 different movies, TV episodes, and shorts with "butterfly effect" in the title, not to mention notable references. in movies like Jurassic Park, or in songs, books and memes.

Oh, memes in pop culture, the butterfly effect has come to mean that even the smallest, seemingly insignificant decisions you make can have huge consequences later in your life and I think the reason people are so fascinated by The butterfly effect is because it reaches a fundamental question: How well can we predict the future? Now the goal of this video is to answer that question by examining the

science

behind the butterfly effect, so if we go back to the late 17th century, after Isaac Newton developed his laws of motion and universal gravitation, everything seemed predictable. I mean we could explain the movements of all the planets and moons, we could predict eclipses and the appearance of comets with millimeter precision centuries in advance.

French physicist Pierre-Simon Laplace summed it up in a famous thought experiment: he imagined a superintelligent being. being, now called Laplace's demon, who knew everything about the current state of the universe: the positions and momenta of all the particles and how they interact, if this intellect were vast enough to subject the data to analysis, he concluded, then the future, like the past, would be present before his eyes. This is total determinism: the view that the future is already set, we just have to wait for it to manifest. I think if you've studied a little bit of physics, this is the From a natural point of view, I mean, I'm sure there is the Heisenberg uncertainty principle of quantum mechanics, but that's on the scale of atoms;

Pretty insignificant on the scale of people. Practically all the problems I studied were those that could be solved analytically, such as the movement of planets, falling objects or pendulums, and speaking of pendulums, I want to see here a case of a simple pendulum to present an important representation of the systems dynamic. which is phase space, so some people may be familiar with position-time or velocity-time graphs, but what if we wanted to make a two-dimensional graph that represents all possible states of the pendulum? As much as possible you could do on a graph, on the x axis we can plot the angle of the pendulum and on the y axis its velocity.

And this is what is called phase space. If the pendulum has friction, it will eventually slow down and stop, and this is shown in phase space by the internal spiral: the pendulum swings slower and less far each time and it doesn't really matter what the initial conditions are, we know that The final state will be the pendulum at rest hanging downwards and from the graph it appears that the system is attracted towards the origin, that fixed point, so it is now called a fixed point attractor if the pendulum does not lose energy, well it swings forward and back in the same way each time and in the phase space we get a loop the pendulum goes faster at the bottom but the oscillation is in opposite directions as it goes back and forth the closed loop tells us that the movement is periodic and predictable every time you see an image like this in phase space, you know that this system repeats regularly, we can swing the pendulum with different amplitudes, but the image in phase space is very similar, just a loop of different size now it is important to note that the curves never intersect in phase space and that is because each point uniquely identifies the entire state of the system and that state has only one future, so once there is the initial state defined, the entire future is determined now, the pendulum can be well understood using Newtonian physics, but Newton himself was aware of problems that did not submit so easily to his equations, in particular the three-body problem . so calculating the movement of the Earth around the Sun was quite simple with just those two bodies, but if one more body was added, say the Moon, it became practically impossible.

Newton told his friend Haley that the theory of the Moon's movements had made him think. The problem, as would become clear to Henri Poincaré two hundred years later, was that there was no simple solution to the three-body problem that Poincaré had envisioned and which would later become the three-body problem. known as

chaos

. The

chaos

really became apparent in the 1960s, when meteorologist Ed Lorenz attempted to make a basic computer simulation of Earth's atmosphere. It had 12 equations and 12 variables, things like temperature, pressure, humidity, etc., and the computer printed every time. I went through like a row of 12 numbers so you could see how they evolved over time.

Now the big breakthrough came when Lorenz wanted to redo a run, but as a shortcut he entered the numbers from half of a previous print and then set up the computer calculating. He went out for coffee, and when he returned and saw the results, Lorenz was stunned. The new series followed the previous one for a short time, but then diverged and very soon described a totally different state of the atmosphere, that is, a totally different climate. Lorenz's first thought, of course, was that the computer had broken. Maybe a vacuum tube. had exploded. But none had done it.

The real reason for the difference came down to the fact that the printer rounded to three decimal places while the computer calculated to six. So when he entered those initial conditions, the difference of less than one part in a thousand created a totally different climate in a short time. Going forward, Lorenz now attempted to simplify his equations and then simplify them some more, down to just three equations and three variables that represented a toy model of convection: essentially a second portion of the atmosphere heated at the bottom and cooled at the bottom. superior, but again. , he got the same kind of behavior: if he changed the numbers just a little bit, the results diverged dramatically.

Lorenz's system showed what is known as sensitive dependence on initial conditions, which is the hallmark of chaos. Now that Lorenz was working with three variables, we can plot the phase space of his system in three dimensions. We can choose any point as our initial state and watch how it evolves. Is our point moving towards a fixed attractor? Or a repeated loop? Not seem. In truth, our system will never revisit the exact same state again. I actually started here with three widely spaced initial states, and they've been evolving together until now, but now they're starting to diverge.

From being arbitrarily together, they end up on totally different trajectories. This is a sensitive dependence on the initial conditions in action. Now I must point out that there is nothing random about this system of equations. It is completely deterministic, just like the pendulum, so if you could input exactly the same initial conditions, you would get exactly the same result. The problem is that, unlike the pendulum, this system is chaotic, so any difference in the initial conditions, no matter how small, is amplified to a totally different final state. It seems like a paradox, but this system is both deterministic and unpredictable because, in practice, you can never know the initial conditions with perfect precision, and I'm talking about infinite decimals.

But the result suggests why even today, with huge supercomputers, it is so difficult to forecast the weather more than a week in advance. In fact, studies have shown that by the eighth day of a long-range forecast, the prediction is less accurate than if we simply took the average historical conditions for that day and, knowing the chaos, meteorologists no longer make a single forecast, but that make ensemble forecasts, varying initial conditions and model parameters to create an ensemble of predictions. Now, far from being the exception to the rule, chaotic systems have been appearing everywhere. The double pendulum, just two simple pendulums connected together, is chaotic.

Here two double pendulums have been released simultaneously with almost the same initial conditions, but no matter how hard you try, you will never be able to release a double pendulum and make it behave the same way twice. . Its movement will always be unpredictable. You might think that chaos always requires a lot of energy or irregular movements, but this system of five fidget spinners with repellent magnets in each of their arms is also chaotic. At first glance, the system appears to repeat itself regularly, but if you look closer, you'll notice some strange movements. A roulette spins suddenly the other way.

Even our solar system is not predictable. A study that simulated our solar system for a hundred million years in the future found that its behavior as a whole is chaotic. with a characteristic time of about four million years, meaning that within, say, 10 or 15 million years, some planets or moons may have collided or been ejected from the solar system entirely. The same system that we consider a model of order is unpredictable even on modest time scales. So how well can we predict the future? Not very well, at least when it comes to chaotic systems. The further into the future you try to predict, the more difficult it becomes, and past a certain point, predictions are no better than guesses.

The same thing happens when we look at the past of chaotic systems and try to identify the initial causes. I think of it as a kind of fog that gets darker the more we try to look into the future or into the past. Chaos puts fundamental limits on what we can do. We can know about the future of the systems and what we can say about their past But there is a positive side Let's look again at the phase space of the Lorenz equations If we start with a bunch of different initial conditions and watch them evolve, initially the motion is messy.

But soon all the points have moved toward or over an object and the object, coincidentally, looks a bit like a butterfly. It is the attractor. For a wide range of initial conditions, the system evolves toward a state on this attractor. Now remember: all the paths drawn here never intersect and never connect to form a loop. If they did, they would continue in that loop forever and the behavior would be periodic and predictable, so each path here is actually an infinite curve in a finite space. But how is it possible? Fractals. But that's a story for another video.

This particular attractor is called the Lorenz attractor. Probably the most famous example of a chaotic attractor, although many others have been found for other systems of equations. Now, if people have heard anything about the butterfly effect, it's usually about how small causes make the future unpredictable, but the

science

behind the butterfly effect also reveals a deep and beautiful structure underlying the dynamics. One that can provide useful information about the behavior of a system. Therefore, you cannot predict how an individual state will evolve, but you can tell how a collection of states evolve and, at least in the case of the Lorenz equations, they take the form of a butterfly.

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