Unstable Systems: The Three-Body Problem and the Butterfly Effect


By Steven Gimble, Ph.D., Gettysburg College

Chaos theory says that knowing the laws of physics does not mean that we can necessarily predict the motion of objects. But can the theory solve the three-body problem and explain the ‘butterfly effect’?

Blue and white trails representing chaotic particle motion. Unstable systems show chaotic motion.
A computer-generated image of the chaotic motion of particles. (Image: Gonin/Shutterstock)

Newton’s Laws and Stable Systems

Image illustrating the elliptical path of a planet around the Sun.
A planet revolving around the Sun. If the Sun is taken to be fixed in place, the planet appears to move in an ellipse around it. (Image: Morphart Creation/Shutterstock)

Newton gave us his three mechanical laws and his gravitational law which said that the attraction between any two masses is inversely proportional to the square of the distance between the objects.

How do two bodies in space interact? How does the gravitational attraction between them make them move? There’s a simple solution to applying Newton’s laws: the bodies move around each other in such a way that if one were thought to be standing still, the other would move around it in an ellipse.

So, a planet such as Earth moves around the Sun in an ellipse.

Learn more about Newton, who inspired the Age of Enlightenment.

The Three-Body Problem and Chaos Theory

But we don’t live in a universe of only two things. Suppose we enlarge our scope to look not just at the Earth and the Sun, but the Earth, the Sun, and the Moon. We have the Moon going around the Earth and the Earth going around the Sun.

The problem is that when we use Newton’s laws for three bodies, the equations are no longer solvable. In the messy, complicated real world, the equations have no exact steady-state solutions: there is no way in traditional mathematical language to describe their path in space.

This is a transcript from the video series Redefining Reality: The Intellectual Implications of Modern Science. Watch it now, on Wondrium.

The important word here is ‘exact’. We don’t have an exact solution, but using simplifications, assumptions, and ignoring certain elements, we can come up with approximations. These approximations are extremely useful and quite good. But while we can get most of the way there, there is much that is interesting in the gap between the approximate and the real solution. It’s in that gap where you find chaos theory.

Stable and Unstable Systems

Scientists love when the state equations for a system are what we call linear. A linear equation can be graphed so that its solutions form a line. A nice property of linear equations is that they are stable; a small change in the initial conditions leads to a correspondingly small change in the outcome.

But in unstable systems, a small difference in initial conditions results in a significant difference in the result. This sensitivity to initial conditions is the hallmark of non-linear equations of unstable systems.

Learn more about chaos theory.

The Butterfly Effect

The commonly used phrase to describe this is the ‘butterfly effect’ which comes from the Ray Bradbury short story ‘A Sound of Thunder’.

In the story, the time machine has been invented and people are transported into the past to hunt dinosaurs. Now, the guides are well aware that the future is contingent on the past in untold ways, so they have constructed a levitating walkway that everyone must stay on and they only hunt dinosaurs that were about to die anyway. Afterward, they remove all bullets, all other traces, leaving the past undisturbed.

During a hunting expedition to shoot a Tyrannosaurus, the customer is so frightened that he falls off the path. The guides do everything they can to minimize the effects of the event, but they return to their original time and find that things have changed. They then find a crushed butterfly under the customer’s boot. The death of that butterfly had effects that changed everything.

Edward Lorenz and the Butterfly Effect

Edward Lorenz, the father of chaos theory, entitled his most famous paper, ‘Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in Texas’? In the 1960s, Lorenz was a meteorologist at MIT where he used one of the first big computers to create the first computer models of the weather.

One day, in the middle of a run, Lorenz had to stop the model. He wrote down the values of all the variables and turned off the machine. When he came back, he restarted the run using the values he had noted down, expecting to see the system develop as before.

Weather is an Unstable System

But Lorenz noticed an incredible thing. The weather in the model that developed was completely different from what was happening when Lorenz stopped the machine. He was puzzled. Could it be an error?

Image illustrating chaotic ,motion in a Lorenz Atrractor. Chaos theory helps show complex motion.
A diagram showing how two different paths (blue and yellow) will be followed by a particle if there is even a very small difference in the starting angle. ( Image: António Miguel de Campos/ Public domain)

Then he realized that he had noted the values of the variables to three decimal places, while the computer computed to six decimal places. Surely, the numbers in the tens of thousandths, the hundreds of thousandths, and the millionths column were not that important?

But he tested it and they were. Incredibly, subtle changes in the values of input into the model yielded massive changes in the weather as it developed. In the first computerized model of the weather, a butterfly flapping its wings in Brazil could generate a tornado in Texas.

Unstable Systems are Sensitive to Initial Conditions

Such sensitivity to initial conditions is a problem because no matter how sensitive your measurements, the act of collecting data itself comes with what scientists call experimental error. By the word ‘error’ here, they do not mean mistakes, but that there’s always a limit to how precisely we can measure something.

But if the system’s sensitivity to initial conditions is sufficiently small, it will far fall below our ability to measure it. The development will appear random and become unpredictable, even though it is thoroughly deterministic.

Chaos theory attempts to show the underlying order in the apparently unpredictable system.

Commonly Asked Questions about the Three-Body problem and the ‘Butterfly Effect’

Q. What is the three-body problem?

The three-body problem refers to the fact that Newton’s laws cannot predict the motion of three bodies interacting with each other. No exact solutions are available to predict the motion of three bodies affected by each other’s gravity.

Q. What is the ‘butterfly effect’?

The ‘butterfly effect’ refers to the fact that, in unstable systems, a small change in initial conditions can make a large difference to the subsequent state of a system.

Q. Why is Edward Lorenz called the ‘Father of Chaos Theory’?

Edward Lorenz was a meteorologist who discovered, in his weather experiments, that a small change in parameters can change the way in which weather develops. This was the beginning of chaos theory.

Q. What is an unstable system in Physics?

An unstable system is a system that shows large differences in later states if the initial conditions are changed even minutely.

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