Chaos Theory: the language of (in)stability

Consider the following situation: You are a computer scientist who specializes in simulations of dynamical systems, such as the weather surrounding your city or the trajectories of multiple planets affected by each other's gravitational pull. This is crucial to your job: you work for weather forecasting and also part-time on an intergalactic intellectual enterprise to attack and colonize the Milky Way. Now, as a result of extensive scientific research and their own ingenuity, their simulations can perfectly predict how these systems change over time, given any set of initial parameters. However, you still have to be very careful when using these simulations to predict what happens in the real world, or else it can result in a wildly inaccurate and completely different prediction.

This is due to the simple fact that these systems are chaotic. These systems that we mentioned above are what is known as chaotic deterministic. Deterministic means that they are, in fact, not random: given the initial conditions, there is one and only one way the system will work. However, the fact that they are chaotic means that they show aperiodic behavior over time (meaning there is no observable pattern to how they behave) and are extremely sensitive to small changes in initial conditions. Of course, it is impossible to discover the exact real-world parameters with zero margin of error, like knowing the temperature of a room to infinite decimal places, and any small differences in initial conditions will gradually turn into larger and larger differences.

As a result, one of the most important tasks when dealing with

chaos

is to figure out: after what period of time is it simply not worth predicting the future? But

chaos

goes far beyond being simply unpredictable: the entire phenomenon and field of study of chaos has its roots in differential equations and dynamical systems, the same

language

used to describe how any physical system in the world evolves. real. This video aims to tell the story of chaos step by step, from simple non-chaotic systems, through different types of attractors, to fractal spaces and the

language

of unpredictability. A dynamic system involves one or more variables that change over time according to autonomous differential equations.

For example, let's say there is a system that has two variables x and y. Point X is the rate of change of x as time changes, and point y is the rate of change of y as time changes; Note that, although it does not show it, x and y depend on the independent variable t, which represents time, and this dot notation is special because it can only be used when the independent variable is time. However, notice how the differential equations describing x dot y y dot do not actually involve t, and only contain x and y as variables.

This makes them autonomous; each combination of x and y only corresponds to a combination of x dot and y dot. As a result, there is a very convenient geometric and visual way to represent a dynamical system, known as phase space. This is a Cartesian space where the axes are the variables of the system. Each point in space is a unique state of the system and has its own rate of change which can be represented as a vector. For this specific system shown, the vector field looks like this. Let's distribute a bunch of random points to represent different possible states and see how they evolve and move.

They all seem to spiral towards the center. This brings us to our next topic. An attractor is a set of points in phase space that attracts all trajectories in a given area surrounding it, known as a basin of attraction. Here, the attractor is simply the origin and the basin of attraction is each point in space. Note that at the origin where x = 0 and y = 0, point x and point y are also equal to 0; that makes it a fixed point, because any point there will stay there forever because it has a rate of change of zero. Since it is also an attractor, it is a fixed point attractor.

Trajectories that are absorbed by any attractor never manage to escape; This seems to reflect some sort of inevitability and predictability that the system will always end up a certain way no matter what. How could this be related to chaos? Well, don't worry, there are also other types of attractors... There was a time when computers were made with vacuum tubes. Balthasar van der Pol worked as an electrical engineer at Philips during the 1920s, and it was while studying vacuum tubes that he stumbled upon a system of differential equations that exhibited interesting behavior and would later become known as the Van der Pol oscillator.

The original equations had a parameter by which x point was multiplied and by which y point was divided, but here the parameter has been chosen to be 1, for a simpler set of equations. If we plot the phase plane of this system, we can see that, interestingly, the trajectories everywhere seem to approach this loop around the origin. This loop is known as a limit cycle attractor and is an important example that shows us that attracting trajectories does not necessarily have to result in trajectories stopping at a singular point. Limit cycles are often characterized by physical phenomena involving some type of oscillation, and van der Pol's equations are no different.

In addition to electrical circuits, they have been used to model things like two tectonic plates on a geological fault line or the mechanics of human vocal cords. But… this is still not completely chaotic. There are two more ingredients. In 1963, meteorologist Edward Lorenz was developing a simulation of atmospheric conditions involving 12 changing variables and discovered that small differences in the initial values ​​soon resulted in disproportionately large differences in the state of the variables. Curious about this, Lorenz spent some time simplifying the simulation of it to have only 3 variables, but still show this sensitive dependence on the initial conditions.

This simplified model describes convection cycles in the atmosphere and is now known as the Lorenz System. It is the symbol of chaos

theory

and is sometimes almost synonymous with the butterfly effect or the field of chaos

theory

itself; It even looks like a butterfly. The Lorenz equations have some parameters that can be modified to alter the behavior of the system, but we will use values ​​of 28, 10, and 8/3. This is what is known as a strange attractor and this is what it means. A strange attractor is one that has a fractal structure. No point in space is visited more than once by the same trajectory;

If that happened, the trajectory would follow a predictable loop. And two paths will never intersect; if that happened, they would merge along the same path, giving the same result to two different sets of initial conditions. Think about what that means: a single trajectory will visit an infinite number of points in this limited space, and this limited space will have an infinite number of trajectories. Now, the paths are just curves, so they should be one-dimensional, right? But how is it that no matter how close you get to this attractor, you can always find more and more trajectories, everywhere?

That is why this attractor is said to have a non-integer dimension: it is formed by infinitely long curves in a finite space, which are so detailed that they begin to partially fill higher dimensions. It is not one-dimensional, two-dimensional or three-dimensional; Its dimension is somewhere in between. As a result of this non-integer dimension and detail at arbitrarily small scales, the set of points in the Lorenz attractor is a fractal space, and so is a strange attractor. A strange attractor is not necessarily chaotic, but a chaotic attractor will always be strange, and the Lorenz attractor is a strange and chaotic attractor.

Notice what happens when I highlight two paths that are initially separated by a very small distance. It's not long until they diverge enough to take completely different paths. It turns out that in the early stages of this divergence, when the two trajectories are close to each other, the distance between them increases exponentially. After a time of t, the resulting difference is the initial difference multiplied by e raised to lambda t (to some extent, of course, since the attractor is limited in size). Here, lambda represents an important value known as the Lyapunov exponent. Since it is a factor of the exponent of e, if it is positive, any distance between trajectories will increase exponentially, if it is equal to zero, the distance will remain constant, and if it is negative, the distance will converge to zero.

There is no way to find the Lyapunov exponent simply by looking at the equations: it is measured by running the simulation, tracking many pairs of trajectories, and finding the average rate of change in their distance. But it provides a simple metric to communicate how chaotic a system is. As long as the Lyapunov exponent is greater than zero, the attractor will be chaotic and is equal to approximately 0.9 for the Lorenz attractor. This is how we can calculate the period of time over which the predictions are valid, also known as the predictability horizon. Rearranging the above equation, we can see that this is how we can find, given the initial error, zero delta, and a maximum error we are willing to allow, a.

In the Lorenz system, after a time of only 10, any error would have been multiplied by more than 8000. To get an idea of ​​how difficult it is to make accurate predictions over long periods of time with this exponential divergence, let's say you have a simulation that predicts where ocean currents flow and you wanted to keep the error below 1000 m. If you ran it twice, once with an initial error of one meter and once with an initial error that was a million times smaller, by one micrometer, how long do you think the simulation with the smaller error would stay below the margin of? mistake?

Well, let's write down the expressions for the two predictability horizons and put them into a fraction, simplify a little, use some logarithm rules and it will be 3 times as long. A million times more accurate and your simulation would be valid for, say, 9 days instead of 3 days. This is the type of difficulty that any chaotic system presents in its simulation. The predictability horizon of the solar system is no more than a few million years with current technology, not even a tenth of the time period between humans and dinosaurs. Earth's atmosphere and climate are incredibly difficult to predict accurately for more than a week.

Maybe people just have to keep tuning into the forecast every day, and it's unlikely that our home will be colonized by an intergalactic alien civilization. That is something good? I'll let you decide. I personally think there is something beautiful about the fact that we can never know exactly what will come in the future. As Master Oogway says: “Yesterday is history, tomorrow is a mystery, but today is a gift. That is why it is called present".