Warning: package 'vembedr' was built under R version 4.4.2Beauty in the Midst of Chaos
Random World
Mathematics
Physics
A flap of a butterfly’s wings in Brazil set off a Tornado in Texas
Towards the Chaotic Phenomenon
In August 2017, people across North America planned for an once-in-a-lifetime opportunity to watch a total solar eclipse. Here’s a reference to the 2017 eclipse in an article about a previous one, 99 years earlier.
For centuries, humans have used math and science to make predictions about our universe. Certainly, by the end of the 19th century, scientists felt that we were living in a Cartesian universe, in which, if you had some omniscient being who knew the positions and velocities of all the particles in the universe, in principle, that being would know what would happen for all time.
But if that’s true, why are some things in our world still so hard to predict, even in an era of supercomputers and big data? If we can predict an eclipse a century in advance, why can we only predict the weather about a week or two in advance?
Where Does the Chaos Originate?
Everything Could be Predictable
In the late 1600s, after Issac Newton had come up with his laws of motion and universal gravitation, everything seemed predictable. We could explain the motions of all the planets and moons, and we could predict an eclipse and the appearance of comets with pinpoint accuracy centuries in advance.
The Past Would be Present Before Its Eyes
French scholar and polymath Pierre-Simon Laplace summed it up in a famous thought experiment; He imagined a super-intelligent being, now called Laplace’s demon, that 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 submit the data to analysis then the future, just like the past would be present before its eyes”
This is total determinism: the view that the future is already fixed, we just have to wait for it to manifest itself. If you have studied a bit of physics, this is the natural viewpoint to come away with. The Heisenberg principle from quantum mechanics, that’s on the scale of atoms; pretty insignificant on the scale of people. Virtually all these problems could be solved analytically, like the motion of planets, falling objects, or pendulums.
In the case of pendulums, one should understand a fact of a simple pendulum which is phase space. It is one of the most important representations of the dynamical system.
Key Concept of Dynamical System
Dynamical systems involve one or more variables that change over time according to autonomous differential equations.
\(\frac{dx}{dt}=\dot{x}=\) rate of change of \(x\) as time changes
\(\frac{dy}{dt}=\dot{y}=\) rate of change of \(y\) as time changes
\(x\) and \(y\) depend on the independent variable \(t\), which stands for time and the dot notation is special in that it can only be used when the independent variable is time.
For example, \(\dot x = -y-0.1x\) and \(\dot y =y-0.4y\) do not involve \(t\) . It only contains \(x\) and \(y\) as variables. This makes them autonomous; each combination of \(x\) and \(y\) only corresponds to one combination \(\dot x\) and \(\dot y\) .
Some people may be familiar with position-time or velocity-time graphs, but what if we wanted to make a 2d plot that represents every possible state of the pendulum?
Phase Space and Simple Pendulums
Every possible thing it could do in one graph, well on the x-axis we can plot the angle of the pendulum, and 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 inward spiral. The pendulum swings slower and less far each time and it does not really matter what the initial conditions are, we know that the final state will be the pendulum at rest hanging straight down, and from the graph, it looks like the system is attracted to the origin, that one fixed point.
Now if the pendulum does not lose energy, well it swings back and forth the same way each time, and in phase space, we get a loop. The pendulum is going the fastest at the bottom but the swing is in opposite directions as it goes back and forth. The close loop tells us the motion is periodic and predictable any time you see an image like this in phase space, you know the system regularly repeats.
Double Pendulum And Phase Space
We can swing the pendulum with different amplitudes, but the picture in phase space is very similar, just a different-sized loop. Now an important thing to note is that the curves never cross in phase space and that’s because each point uniquely identifies the completion of the system and that state has only one future. So once you’ve defined the initial state, the entire future is determined. This is what is called “Deterministic Chaos”. Deterministic, because its future is determined by physical laws, although it seems random.
Now the mechanics of a pendulum can be well understood using Newtonian physics, but Newton himself was aware of problems that did not submit to his equations so easily, particularly the three-body problem. So calculating the motion of the Earth around the Sun was simple enough with just those two bodies, but add in one more, say the moon, and it became virtually impossible. Newton told his friend Edmond Halley that the theory of the motions of the moon made him a headache, and kept him awake so often that he would think it no more.
Poincare’s Research
In 1903, French mathematician Henri Poincare observed something similar while studying the problem of three objects in space, all affected by each other’s gravity. He was always looking for what’s the most simple set of equations that exhibit chaotic properties. The problem, as would become clear to Henri Poincare two hundred years later, was that there was no simple solution to the three-body problem.
The Motivation of Chaos
Consider the following situations: you are a computer scientist specializing in simulations of dynamical systems such as the weather surrounding your city, or the trajectories of multiple planets affected by each other’s gravitational pulls. This is crucial to your job- you work for the weather forecast, and also part-timer in an intergalactic mastermind’s venture to attack and colonize the Milky Way. Now, as a result of extensive scientific research and your own ingenuity, your simulations can perfectly predict how these systems change over time, given any set of starting parameters. However, you must still be extremely careful in using these simulations to predict what happens in the real world, or else it may result in a widely inaccurate, completely different prediction. This is due to the simple fact that these systems, are chaotic.
Definition of Dynamical System
A dynamical system is a system that has a function that describes the time dependence of a point in a geometrical space. More formally, a dynamical system is defined as a “particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives”. To predict the system’s future behavior, an analytical solution of such equations or their integration over time through computer simulation is made.
The study of dynamical systems is the focus of dynamical systems theory, which has applications in a wide variety of fields such as mathematics, physics, engineering, biology, chemistry, economics, medicine, and history. They are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and also the edge of chaos concept.
Pioneer of Chaos
Chaos came into focus in the 1960s, when meteorologist Edward Norton Lorenz tried to make a basic computer simulation of the Earth’s atmosphere. He had 12 equations and 12 variables, things like temperature, pressure, humidity, and so on. And the computer would print out each time step as a row of 12 numbers. So you could watch how they evolved over time.
Now the breakthrough came when Lorenz wanted to redo a run but as a shortcut, he entered the numbers from halfway through a previous printout and then he set the computer to calculate. After a while when he saw the results, Lorenz was stunned.
The new run followed the old one for a short while but then it diverged and pretty soon it was describing a different state of the atmosphere (different weather). Lorenz’s first thought, of course, was that the computer had broken (Maybe the Vacuum tube had blown, but none had ). The real reason for the difference came down to the fact the printer rounded to three decimal places whereas the computer calculated with six.
So, when he entered those initial conditions, the difference of less than one part in a thousand created different weather just a short time into the future. Now, Lorenz tried simplifying his equations and then simplifying them some more, down to just three equations and three variables that represent the toy model of convection:
\[\dfrac{dx}{dt}=\sigma(y-x)\]
\[ \dfrac{dy}{dt}=x(\rho-z)-y \]
\[ \dfrac{dz}{dt}=xy-\beta z \]
essentially a 2d slice of the atmosphere heated at the bottom and cooled at the top. But again, he got the same type of behavior; if he changed the numbers just a tiny bit, results diverged dramatically.
Lorenz’s System
Lorenz’s system displayed what’s become known as the sensitive dependencies on initial time conditions, which is the hallmark of chaos. Today we can create models of the atmosphere that go far beyond Lorenz’s columns of numbers.
In this model from MIT, the white line represent a bunch of balloons released from approximately the same point. Eventually, the Balloons diverge along very different paths, because of the small differences in their starting points. If you change the initial conditions ever so slightly, infinitesimally slightly. The two trajectories seem to go along with each other and then they diverge exponentially fast. So, that is chaos. It means that you have to know a system with infinite precision to be able to predict it infinitely in time.
Sensitive Dependence on Initial Conditions
When Lorenz found a set of three equations which is the simplified version of equations used to model convection, with only three variables that change over time. It is notable for having chaotic solutions for certain parameter values and initial conditions.\[\dfrac{dx}{dt}=\sigma(y-x)\]
\[\dfrac{dy}{dt}=x(\rho-z)-y \]
\[\dfrac{dz}{dt}=xy-\beta z\]
with the initial conditions : \(x(0)=y(0)=z(0)=1\)
The equations relate the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. In particular, the equations describe the rate of change of three quantities with respect to time: \(x\) is proportional to the rate of convection, \(y\) to the horizontal temperature variation, and \(z\) to the vertical temperature variation. The constants \(\sigma\), \(\rho\), and \(\beta\) are system parameters proportional to the Prandtl number, Rayleigh number, and certain physical dimensions of the layer itself where \(\sigma\), \(\rho\), and \(\beta\) are three parameters, with values of \(\sigma = 10,\ \beta= \frac{8}{3}, \rho=28\).
Since Lorenz was working with three variables, we can plot the phase space of his system in three dimensions. We can pick any points as our initial state and watch how it evolves.