The phrase has strong ties to the research of meteorologist and mathematician Edward Norton Lorenz. He explained that the metaphorical example of a tornado's specifics (formation time, path) being impacted by minute disturbances like a far-off butterfly fluttering its wings weeks beforehand is where the butterfly effect originated.

By 1972, Lorenz had been convinced to add a butterfly and tornado to make it seem more poetic, having previously used a seagull generating a storm. He noticed the effect when he saw seemingly insignificant rounding of beginning condition data in runs of his weather model. He pointed out that when using unrounded initial condition data, the weather model would not be able to replicate the outcomes of runs. An extremely slight alteration to the starting conditions had produced a totally different result. Since then, the idea of the "butterfly effect" has been used broadly to any circumstance in which a minor alteration is thought to have a greater impact than previously, even beyond the field of meteorology.

According to Lorenz, Philip Merilees came up with the subject Does the flap of a butterfly's wings in Brazil spark off a tornado in Texas? for a talk he was scheduled to present at the 139th meeting of the American Association for the Advancement of Science in 1972 as the heading. While the image of a butterfly flapping its wings has stayed consistent in representing this idea, there have been significant variations in the butterfly's placement, the consequences, and the location of the repercussions.

The expression alludes to the theory that a butterfly's wings might cause minute modifications in the atmosphere, which could eventually affect a tornado's course or delay, speed, or even stop a tornado from occurring in another area. Although the butterfly does not directly cause or power a tornado, the term is meant to suggest that the butterfly's wing flaps may do so. This is because the butterfly's wing flaps are one of the initial conditions of a complex web that is interconnected; one set of conditions can lead to a tornado, while another set of conditions cannot. The flying wing symbolises a tiny modification to the system's starting conditions that triggers significant changes to the course of events (compare: domino effect). The system's course might have been very different if the butterfly hadn't fluttered its wings, but it's also possible that the circumstances that lead to a tornado would have existed even if the butterfly hadn't flapped its wings.

Since it is impossible to know with absolute certainty what will happen at the beginning of a system, like the weather, the butterfly effect poses a clear prediction difficulty. The creation of ensemble forecasting, which makes many forecasts given altered beginning conditions, was spurred by this issue.

Although the term "butterfly effect" is frequently used to refer to sensitive dependence on initial conditions of the type that Lorenz described in his 1963 paper (and that Poincaré had previously observed), the metaphor originated with work he published in 1969 that advanced the concept. Lorenz put forth a mathematical theory explaining how minute movements in the atmosphere have an impact on bigger systems. He discovered that, as long as the error is not zero, reducing the error in the initial conditions will not increase the predictability of the systems in that model beyond a certain point in the future. This proved that, in terms of predictability, a deterministic system could be "observationally indistinguishable" from a non-deterministic one. Comparable to the issues posed by quantum physics, recent reexaminations of this study indicate that it presented a serious challenge to the notion that our universe is deterministic.

In physical systems

In weather

The butterfly effect is most commonly associated with weather; typical weather prediction models, for instance, provide an easy way to illustrate it. The development of weather prediction techniques relies heavily on chaos, as models are susceptible to beginning conditions, as explained by climate scientists James Annan and William Connolley. The disclaimer is added: "We have many more pressing uncertainties to be concerned about, so of course the existence of an unknown butterfly fluttering its wings has no direct influence on weather forecasts, since it will take far too long for such a little disruption to build to a considerable scale.

Therefore, this phenomenon's direct influence on weather forecast is frequently erroneous." There are some differences between the two types of butterfly effects, such as the sensitive reliance on starting conditions and the capacity of a little disturbance to produce a well-organized circulation over great distances. It has been recorded to compare the two types of butterfly effects and the third type of butterfly effect. According to recent research, both meteorological and non-meteorological linear models have demonstrated that instability contributes to the butterfly effect, which is characterised by a modest disruption creating a brief but considerable exponential expansion.

In quantum mechanics

In semiclassical and quantum physics, the butterfly effect—a sensitive dependence on initial conditions—has been examined in several scenarios, including as atoms under strong fields and the anisotropic Kepler problem. Although some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments, Martin Gutzwiller and John B. Delos and colleagues' semiclassical treatments do include the sensitive dependence on initial conditions shown in classical motion. Certain forms of the butterfly effect in quantum physics are shown to be nonexistent by simulations using quantum computers and random matrix theory.

It has been suggested by other authors that quantum systems exhibit the butterfly effect. In this study, Zbyszek P. Karkuszewski and colleagues examine the evolution of quantum systems with marginally distinct Hamiltonians across time. They look at how sensitive quantum systems are to even minute modifications to the Hamiltonians that they are given. A quantum algorithm that "measures the rate at which identical initial states diverge when subjected to slightly different dynamics" was introduced by David Poulin et al. to quantify fidelity decay. "The closest quantum analogue to the (purely classical) butterfly effect" is how they characterise fidelity loss.

The quantum butterfly effect takes into account the impact of a tiny change in the Hamiltonian system with a given initial location and velocity, in contrast to the classical butterfly effect, which takes into account the impact of a small change in the position and/or velocity of an object in a given Hamiltonian system. Experimental evidence for this quantum butterfly phenomenon exists. Quantum chaos refers to quantum and semiclassical approaches to system sensitivity to beginning conditions.

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