The difficulty in predicting what happens in chaotic systems comes from how minute differences in inputs can become dramatic changes to the output, in what’s known as sensitivity to initial conditions. This is not an issue in classical Newtonian physics, where the regular movement of objects — planetary orbits, swinging pendulums, rolling balls — are easily predicted, even allowing for small changes to inputs.

Sensitivity to initial conditions is also known more commonly as the “butterfly effect,” which suggests the extreme possibility that a butterfly flapping its wings in the Amazonian jungle might cause a storm to rage across Europe some weeks later. I’m reminded of the idea every time I play a game of pool. What ostensibly appears to be a classical system — balls whizzing around the table — is more akin to a chaotic one. On rare occasions, my break-off shot is near-perfect, yet it is utterly impossible to replicate. One could find equations to model the movement of balls around the table, factoring in things such as their mass or the exact strength with which I strike the cue ball. But the slightest deviation in any of these conditions results in wildly different outcomes.

The idea is a popular theme in literature. Ray Bradbury’s 1952 short story “A Sound of Thunder” speculates on how the single action of a time traveler might alter the course of history in disconcerting ways (for instance, accidentally stomping on a golden butterfly 65 million years ago results in a significant change to language in the present, with people now uttering words phonetically). 

More famously, chaos theory is the central theme of Michael Crichton’s Jurassic Park, published in 1990 when the field was entering the mainstream. In the novel, mathematician Ian Malcolm prophetically warns John Hammond, CEO of the bioengineering company InGen, of the dangers of using genetically recreated dinosaurs as the basis for an amusement park. Hammond’s hope is that he can tame the beasts and keep them safely locked up. Malcolm, a specialist in chaos theory, argues that this is futile, drawing on the key lessons of his field:

Chaos theory teaches us … that straight linearity, which we have come to take for granted in everything from physics to fiction, simply does not exist. … Real life isn’t a series of interconnected events occurring one after another like beads strung on a necklace. Life is actually a series of encounters in which one event may change those that follow in a wholly unpredictable, even devastating way.

It must be said that the film adaptation does not quite manage this degree of nuance. Viewers also miss out on Crichton’s use of the Heighway fractal, a dragon-like curve that gradually ] morphs from a single line into something far more daunting. Crichton includes successive iterations of the fractal at the start of each chapter, a visual metaphor of how the simplest things can spiral out of control.

The butterfly effect reinforces the counterintuitive idea also present in Tolstoy’s reading of history that our lives are determined not by the big but by the vanishingly small. When the cognitive psychologist Amos Tversky was asked what got him into the field, his response sounded like that of a chaos theorist. “It’s hard to know how people select a course in life,” said Tversky. “The big choices we make are practically random. The small choices probably tell us more about who we are. Which field we go into may depend on which high school teacher we happen to meet. Who we marry may depend on who happens to be around at the right time of life.”

In fact, chaos theory goes further, suggesting that those small causal factors may be impossible to determine. As I write this, I have a cast on one wrist and a splint on another, having been diagnosed with a scaphoid fracture on both following an eminently avoidable bike accident. The most basic explanation for my fall is that I misjudged a sharp turn at a roadblock and, with one hand gripping a snooker cue case, I was unable to recover my balance with my other hand. There is no excusing my recklessness in cycling without a firm grasp of both handlebars. But what possessed me to carry the cue in the first place? It was a weekday, so my work schedule must have allowed a 90-minute slot — a canceled meeting, perhaps. Or maybe my poor snooker form in a previous session left me with an itch to get back on the practice table. The car must have been unavailable because otherwise I would have combined the trip with a visit to the swimming pool. The most chaotic system of all — the weather — was a key determinant: If it had rained that day, I may have gone for a run instead. And if the roadworks had started a day later then, well …

Our lives unfold in ways we cannot fully comprehend. The only thing we can be sure of, with mathematical certainty, is that the triumphs we celebrate may have tilted toward tragedy but for the proverbial flapping of a butterfly’s wing.

The central lesson of chaos theory is not that small actions escalate into larger ones — after all, not every action results in a broken wrist or a dinosaur stampede — but that our actions exist alongside many others in a system so complex that the effects of individual actions, small or large, are impossible to determine. On some level, we all have to plead ignorance to the true causes of everyday events.

We make sense of the good and bad in life by assigning particular causes and ignoring the hidden minutiae of everyday events. Chaos theory counsels us to think twice before assigning blame, or credit, for a given situation. It also cautions us against judging others. Even if our complex personalities can be reduced to simple governing principles, there’s no real knowing what “initial conditions” led us to be how we are. This is not to let humans off the hook for bad behavior, but simply to recognize the challenge inherent in empathy. If we wish to walk in another person’s shoes, we had better be prepared to venture deep into their history in our search for root causes. Even then, we must accept that the answers may be elusive. Our lives unfold in ways we cannot fully comprehend. The only thing we can be sure of, with mathematical certainty, is that the triumphs we celebrate may have tilted toward tragedy but for the proverbial flapping of a butterfly’s wing.

Chaos theory has never felt more relevant to our interpretation of everyday events. As the world becomes ever more interconnected, it also becomes more sensitive to our individual deeds. 

As the political scientist Brian Klaas puts it: ‘We control nothing, but influence everything.”

This strikes me as a hopeful realization; we all impact the world in ways we probably fail to appreciate. It is also a daunting one because with such power comes responsibility. The words we utter, the actions we take — even the seemingly mundane ones like the way we engage with social media posts we disagree with — have the potential to morph into events that lie beyond the event horizon of predictability, affecting others to a degree we are unable to fathom. We should exercise them with care.