It's a normal day, and you are roaming in your garden when you spot a tiny beautiful butterfly. You are amazed by how this small creature is so determined to its work, and how it flaps its wings.

After some time, when you reach home, you spend your day normally, not knowing that because of that butterfly's movement, a ripple effect start, and the small movements of the butterfly wing actually result in tornados in some nearby country.

Many of us will consider it a sci-fi thing, but it is quite interesting to know that it might happen in reality.

Let's learn a bit more about the Butterfly Effect.

What is Butterfly Effect?
In Chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.

The work is closely associated with the work of mathematician and meteorologist Edward Lorenz. He noted that the butterfly effect is derived from the metaphorical example of the details of a tornado (the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distant butterfly flapping its wings several weeks earlier. Lorenz discovered the effect when he observed runs of his weather model with initial condition data that were rounded in a seemingly inconsequential manner. He noted that the weather model would fail to reproduce the results of runs with the ungrounded initial condition data. A very small change in initial conditions had created a significantly different outcome.

The idea that small causes may have large effects on the weather was earlier recognized by French mathematician and engineer Henri Poincare. American mathematician and philosopher Norbert Wiener also contributed to this theory. Edward Lorenz's work placed the concept of instability of the Earth's atmosphere onto a quantitative base and linked the concept of instability to the properties of large classes of dynamic systems which are undergoing nonlinear dynamics and deterministic chaos.

The butterfly effect concept has since been used outside the context of weather science as a broad term for any situation where a small change is supposed to be the cause of larger consequences.

History
In The Vocation of Man (1800), Johann Gottlieb Fichte says "you could not remove a single grain of sand from its place without thereby ... changing something throughout all parts of the immeasurable whole".

Chaos theory and the sensitive dependence on initial conditions were described in numerous forms of literature. This is evidence by the case of the three-body problem by Henry Poincare in 1890. He later proposed that such phenomena could be common, for example, in meteorology.

In 1898, Jacques Hadamard noted a general divergence of trajectories in space of negative curvature. Pierre Duhem discussed the possible general significance of this in 1908.

The idea that the death of one butterfly could eventually have a far-reaching ripple effect on subsequent historical events made its earliest known appearance in "A sound of thunder", a 1952 short story by Ray Bradbury. "A Sound of Thunder" discussed the probability of time travel.

In Physical Systems
Weather: The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction models, for example. The climate scientists James Annan and William Connolley explain that chaos is important in the development of weather prediction methods; models are sensitive to initial conditions. They add the caveat: "Of course the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about. So the direct impact of this phenomenon on weather prediction is often somewhat wrong.

Quantum Mechanics: The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases in semiclassical and quantum physics including atoms in strong fields and the anisotropic Kepler problem. Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments; however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by Martin Gutzwiller and Delos and co-workers. The random matrix theory and simulations with quantum computers prove that some versions of the butterfly effect in quantum mechanics do not exist.

Other authors suggest that the butterfly effect can be observed in quantum systems. Karkuszewski et al consider the time evolution of quantum systems that have slightly different Hamiltonians. They investigate the level of sensitivity of quantum systems to small changes in their given Hamiltonians. Poulin et al presented a quantum algorithm to measure fidelity decay, which "measures the rate at which identical initial states diverge when subjected to slightly different dynamics". They consider fidelity decay to be "the closest quantum analog to the (purely classical) butterfly effect". Whereas the classical butterfly effect considers the effect of small change in the position and/or velocity of an object in a given Hamiltonian system, the quantum butterfly effect considers the effect of a small change in the Hamiltonian system with a given initial position and velocity. This quantum butterfly effect has been demonstrated experimentally. Quantum and semiclassical treatments of system sensitivity to initial conditions are known as quantum chaos.

Reference:
1. The Butterfly Effect retrieved from https://en.wikipedia.org/wiki/Butterfly_effect