Chaos: The Science of the Butterfly Effect
The butterfly effect is the idea that tiny causes, like the flap of a butterfly’s wings in Brazil, can have huge effects, like setting off a tornado in Texas. That idea comes straight from the title of a scientific paper published nearly 50 years ago and has perhaps captured the public imagination more than any other recent scientific concept. I mean, on IMDB there are not one but 61 different movies, TV episodes, and short films with ‘butterfly effect’ in the title, not to mention prominent references in movies like Jurassic Park, or in songs, books, and memes.
In pop culture, the butterfly effect has come to mean that even tiny, seemingly insignificant choices you make can have huge consequences later on in your life. And I think the reason people are so fascinated by the butterfly effect is because it gets at a fundamental question: how well can we predict the future?
To answer that question, we need to examine the science behind the butterfly effect.
So if you go back to the late 1600s, after Isaac 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, predict eclipses and the appearances of comets with pinpoint accuracy centuries in advance. French physicist 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, he concluded, 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.
I think if you’ve studied a bit of physics, this is the natural viewpoint to come away with. Sure, there’s Heisenberg’s uncertainty principle from quantum mechanics, but that’s on the scale of atoms; pretty insignificant on the scale of people. Virtually all the problems I studied were ones that could be solved analytically, like the motion of planets, or falling objects, or pendulums.
Speaking of pendulums, I want to look at a case of a simple pendulum here to introduce an important representation of dynamical systems, which is phase space.
Phase Space
Some are familiar with position-time or velocity-time graphs. But what if we wanted to make a 2D plot representing every possible state of the pendulum? On the x-axis, we can plot the angle of the pendulum, and on the y-axis, its velocity. This is phase space. If the pendulum has friction, it eventually slows and stops, shown in phase space by the inward spiral — the pendulum swings slower and less far each time. Regardless of initial conditions, the final state is the pendulum at rest, hanging straight down. The graph looks like the system is attracted to the origin, a fixed point attractor.
If the pendulum doesn’t lose energy, it swings back and forth consistently. In phase space, we get a loop. The pendulum moves fastest at the bottom, with the swing in opposite directions as it goes back and forth. The closed loop indicates periodic and predictable motion. Swinging the pendulum with different amplitudes shows similar phase space loops, just different-sized ones.
Notably, curves never cross in phase space because each point uniquely identifies the complete state of the system, with only one future. Once you’ve defined the initial state, the entire future is determined. The pendulum can be well understood using Newtonian physics, but Newton himself was aware of problems resistant to his equations, particularly the three-body problem. Calculating the Earth’s motion around the Sun was simple enough with just two bodies. But adding one more, say the moon, made it virtually impossible. Newton admitted that the theory of the moon’s motions gave him such headaches that he would think of it no more. The issue, as Henri Poincaré would clarify two centuries later, was the absence of a simple solution to the three-body problem. Poincaré glimpsed what later became known as chaos.
Chaos
Chaos really emerged in the 1960s, when meteorologist Ed Lorenz tried to make a basic computer simulation of the Earth’s atmosphere. He used 12 equations and 12 variables — temperature, pressure, humidity, etc. — and the computer printed out each time step as a row of 12 numbers to track how they evolved over time. The breakthrough came when Lorenz wanted to redo a run but took a shortcut by entering numbers from halfway through a previous printout, then set the computer calculating. When he returned and saw the results, Lorenz was stunned. The new run initially followed the old one, but soon diverged, leading to a totally different atmospheric state — completely different weather. Lorenz’s first thought was a computer malfunction, but none had occurred. The difference came down to the printer rounding to three decimal places, while the computer calculated with six. The difference of less than one part in a thousand created totally different weather shortly into the future.
Lorenz simplified his equations to just three equation and three variables, representing a toy model of convection: a 2D slice of the atmosphere heated at the bottom and cooled at the top. Again, he saw the same behavior: tiny changes in numbers led to dramatically different results. Lorenz’s system displayed what’s known as sensitive dependence on initial conditions, the hallmark of chaos.
Sensitive Dependence
Since Lorenz worked with three variables, we can plot his system’s phase space in three dimensions. Picking any point as our initial state, we observe its evolution. Does it move toward a fixed attractor or a repeating loop? It doesn’t. In reality, the system never revisits the exact same state. Starting with three closely spaced initial states, they evolve together for a while, then diverge, ending on different trajectories. This is sensitive to dependence on initial conditions in action.
Though deterministic, unlike the pendulum, this system is chaotic, so any tiny difference in initial conditions will lead to a completely different final state. It’s both deterministic and unpredictable. Because in practice, you could never know initial conditions with perfect accuracy — infinite decimal places. This suggests why forecasting weather more than a week in advance remains challenging, despite supercomputers. Studies show that by the eighth day of a long-range forecast, predictions are less accurate than using historical average conditions for that day. Knowing about chaos, meteorologists no longer make single forecasts but ensemble forecasts, varying initial conditions and model parameters to create a set of predictions.
Chaos Everywhere
Chaotic systems are not rare. The double pendulum — two simple pendulums connected — is chaotic. Two double pendulums released simultaneously with almost the same initial conditions will diverge in motion, never behaving identically twice. Even a low-energy system like five fidget spinners with repelling magnets in each arm can be chaotic, displaying irregular motions despite seeming regular.
Even our solar system is unpredictable. A study simulating it for a hundred million years found its behavior chaotic, with a characteristic time of about four million years. Within 10 to 15 million years, some planets or moons may collide or be ejected from the solar system. The very system we think of as the model of order is unpredictable on even modest timescales.
Predicting the Future
So, how well can we predict the future? Not very well, at least for chaotic systems. The further into the future you try to predict, the harder it becomes, eventually turning predictions into mere guesses. The same is true when looking into the past of chaotic systems to identify initial causes. It’s like a fog setting in the further we look into the future or past. Chaos limits what we can know about the future of systems and what we can say about their past.
The Silver Lining
However, there is a silver lining. Consider the phase space of Lorenz’s simplified model. For any initial condition, the system doesn’t wander arbitrarily but stays within a well-defined subset of phase space, known as the Lorenz attractor. Instead of fixed-point or periodic attractors, Lorenz discovered the first chaotic attractor, a fractal. Its complexity and fine structure make exact state prediction impossible in the distant future, but its overall shape provides some predictive power. The system will always remain within the attractor’s shape.
So to summarize, while the idea of the butterfly effect suggests a kind of mysterious, almost magical connection between small events and massive consequences, the scientific reality behind it reveals a world of intricate, deterministic yet unpredictable systems. Understanding chaos and sensitive dependence on initial conditions challenges our perceptions of predictability, but it also provides us with a deeper appreciation of the complex and beautiful patterns that arise from the simplest of rules.
