"Inchaos theory, thebutterfly effectis the sensitive dependence oninitial conditionsin which a small change in one state of adeterministicnonlinear systemcan result in large differences in a later state.

The term, closely associated with the work ofEdward Lorenz, 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 the flapping of the wings of a distantbutterflyseveral weeks earlier. Lorenz discovered the effect when he observed that runs of hisweather modelwith initial condition data that were rounded in a seemingly inconsequential manner would fail to reproduce the results of runs with the unrounded 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 in general and in weather specifically was earlier recognized by French mathematician and engineerHenri Poincarand American mathematician and philosopherNorbert Wiener.Edward Lorenz's work placed the concept ofinstabilityof the Earth'satmosphereonto a quantitative base and linked the concept of instability to the properties of large classes of dynamic systems which are undergoingnonlinear dynamicsanddeterministic chaos."

Theory and mathematical definition:

Recurrence, the approximate return of a system towards its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of makingcomplex systems, such as theweather, difficult to predict past a certain time range (approximately a week in the case of weather) since it is impossible to measure the starting atmospheric conditions completely accurately.

In weather

The butterfly effect is most familiar in terms ofweather; 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."

In quantum mechanics:

The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases insemiclassicalandquantum physicsincluding atoms in strong fields and the anisotropicKepler 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 byMartin Gutzwillerand Delos and co-workers.

Other authors suggest that the butterfly effect can be observed in quantum systems. Karkuszewski et al. consider the time evolution of quantum systems which have slightly differentHamiltonians. 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 a small change in the position and/or velocity of an object in a givenHamiltonian 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 asquantum chaos

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