1, 2), meaning that, like a clock, small perturbations do not appreciably influence its dynamics? But, if all that is available are measurements of how a real, non-simulated system evolves over time-for e.g., how a neuron’s membrane potential changes over time, or how the brightness of a star changes over time-how can it be determined if those observations come from a chaotic system? Or if they are just noise? Or if the system is in fact periodic (Glossary, Supplementary Figs. It is relatively simple to determine if a simulated system is chaotic: just run the simulation a few times, with very slightly different initial conditions, and see how quickly the simulations diverge (Supplementary Fig.

These are only a few examples of the many places in nature where chaos has been found. This conceptually simple phenomenon-i.e., extreme sensitivity to small perturbations-is thought to appear everywhere in nature, from cosmic inflation 1, to the orbit of Hyperion 2, to the Belousov–Zhabotinskii chemical reaction 3, to the electrodynamics of the stimulated squid giant axon 4. The meteorologist Edward Lorenz famously described this phenomenon as the butterfly effect: in a chaotic system, something as small as the flapping of a butterfly’s wings can cause an effect as big as a tornado. Formally, a system is chaotic if it is bounded (meaning that, like a planet circling a star, its dynamics stay inside an orbit rather than escaping off to infinity), and if it is deterministic (meaning that, with the exact same initial conditions, it will always evolve over time in the same way), and if tiny perturbations to the system get exponentially amplified (Glossary (Supplementary Information), Supplementary Figs.

A remarkable diversity of natural phenomena are thought to be chaotic.

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