Butterfly effect

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 evidenced by the case of the three-body problem by Poincaré in 1890. He later proposed that such phenomena could be common, for example, in meteorology. In 1898, Jacques Hadamard noted general divergence of trajectories in spaces 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 in which a time traveller alters the future by inadvertently treading on a butterfly in the past. More precisely, though, almost the exact idea and the exact phrasing —of a tiny insect's wing affecting the entire atmosphere's winds— was published in a children's book which became extremely successful and well-known globally in 1962, the year before Lorenz published: In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario. Lorenz wrote: In 1963, Lorenz published a theoretical study of this effect in a highly cited, seminal paper called Deterministic Nonperiodic Flow (the calculations were performed on a Royal McBee LGP-30 computer). The phrase refers to the effect of a butterfly's wings creating tiny changes in the atmosphere that may ultimately alter the path of a tornado or delay, accelerate, or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado, but the term is intended to imply that the flap of the butterfly's wings can cause the tornado: in the sense that the flap of the wings is a part of the initial conditions of an interconnected complex web; one set of conditions leads to a tornado, while the other set of conditions doesn't. The flapping wing creates a small change in the initial condition of the system, which cascades to large-scale alterations of events (compare: domino effect). Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different—but it's also equally possible that the set of conditions without the butterfly flapping its wings is the set that leads to a tornado. The butterfly effect presents an obvious challenge to prediction, since initial conditions for a system such as the weather can never be known to complete accuracy. This problem motivated the development of ensemble forecasting, in which a number of forecasts are made from perturbed initial conditions. Some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed. David Orrell argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role. Stephen Wolfram also notes that the Lorenz equations are highly simplified and do not contain terms that represent viscous effects; he believes that these terms would tend to damp out small perturbations. Recent studies using generalized Lorenz models that included additional dissipative terms and nonlinearity suggested that a larger heating parameter is required for the onset of chaos. While the "butterfly effect" is often explained as being synonymous with sensitive dependence on initial conditions of the kind described by Lorenz in his 1963 paper (and previously observed by Poincaré), the butterfly metaphor was originally applied which took the idea a step further. Lorenz proposed a mathematical model for how tiny motions in the atmosphere scale up to affect larger systems. He found that the systems in that model could only be predicted up to a specific point in the future, and beyond that, reducing the error in the initial conditions would not increase the predictability (as long as the error is not zero). This demonstrated that a deterministic system could be "observationally indistinguishable" from a non-deterministic one in terms of predictability. Recent re-examinations of this paper suggest that it offered a significant challenge to the idea that our universe is deterministic, comparable to the challenges offered by quantum physics. In the book entitled The Essence of Chaos published in 1993, ==Illustrations==

: ==Theory and mathematical definition==

for values ρ=28, σ = 10, β = 8/3. The butterfly effect or sensitive dependence on initial conditions is the property of a dynamical system that, starting from any of various arbitrarily close alternative initial conditions on the attractor, the iterated points will become arbitrarily spread out from each other. Recurrence, the approximate return of a system toward its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, 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. A dynamical system displays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical. Lorenz defined sensitive dependence as follows: The property characterizing an orbit (i.e., a solution) if most other orbits that pass close to it at some point do not remain close to it as time advances. If M is the state space for the map f^t, then f^t displays sensitive dependence to initial conditions if for any x in M and any δ > 0, there are y in M, with distance such that 0 and such that :d(f^\tau(x), f^\tau(y)) > \mathrm{e}^{a\tau} \, d(x,y) for some positive parameter a. The definition does not require that all points from a neighborhood separate from the base point x, but it requires one positive Lyapunov exponent. In addition to a positive Lyapunov exponent, boundedness is another major feature within chaotic systems. The simplest mathematical framework exhibiting sensitive dependence on initial conditions is provided by a particular parametrization of the logistic map: :x_{n+1} = 4 x_n (1-x_n) , \quad 0\leq x_0\leq 1, which, unlike most chaotic maps, has a closed-form solution: :x_{n}=\sin^{2}(2^{n} \theta \pi) where the initial condition parameter \theta is given by \theta = \tfrac{1}{\pi}\sin^{-1}(x_0^{1/2}). For rational \theta, after a finite number of iterations x_n maps into a periodic sequence. But almost all \theta are irrational, and, for irrational \theta, x_n never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2n shows the exponential growth of stretching, which results in sensitive dependence on initial conditions (the butterfly effect), while the squared sine function keeps x_n folded within the range [0, 1]. ==In physical systems==

In weather Overview 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." Differentiating types of butterfly effects The concept of the butterfly effect encompasses several phenomena. The two kinds of butterfly effects, including the sensitive dependence on initial conditions, In Palmer et al., A comparison of the two kinds of butterfly effects In recent studies, it was reported that both meteorological and non-meteorological linear models have shown that instability plays a role in producing a butterfly effect, which is characterized by brief but significant exponential growth resulting from a small disturbance. Recent debates on butterfly effects The first kind of butterfly effect (BE1), known as SDIC (Sensitive Dependence on Initial Conditions), is widely recognized and demonstrated through idealized chaotic models. However, opinions differ regarding the second kind of butterfly effect, specifically the impact of a butterfly flapping its wings on tornado formation, as indicated in two 2024 articles. In more recent discussions published by Physics Today, it is acknowledged that the second kind of butterfly effect (BE2) has never been rigorously verified using a realistic weather model. While the studies suggest that BE2 is unlikely in the real atmosphere, For the third kind of butterfly effect, the limited predictability within the Lorenz 1969 model is explained by scale interactions in one article the presence of SDIC (commonly known as the butterfly effect) implies that chaotic systems have a finite predictability limit. In a literature review, it was found that Lorenz's perspective on the predictability limit can be condensed into the following statement: • (A). The Lorenz 1963 model qualitatively revealed the essence of a finite predictability within a chaotic system such as the atmosphere. However, it did not determine a precise limit for the predictability of the atmosphere. • (B). In the 1960s, the two-week predictability limit was originally estimated based on a doubling time of five days in real-world models. Since then, this finding has been documented in Charney et al. (1966) and has become a consensus. Recently, a short video has been created to present Lorenz's perspective on predictability limit. A recent study refers to the two-week predictability limit, initially calculated in the 1960s with the Mintz-Arakawa model's five-day doubling time, as the "Predictability Limit Hypothesis." Inspired by Moore's Law, this term acknowledges the collaborative contributions of Lorenz, Mintz, and Arakawa under Charney's leadership. The hypothesis supports the investigation into extended-range predictions using both partial differential equation (PDE)-based physics methods and Artificial Intelligence (AI) techniques. In 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 John B. 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. Zbyszek P. Karkuszewski et al. consider the time evolution of quantum systems which have slightly different Hamiltonians. They investigate the level of sensitivity of quantum systems to small changes in their given Hamiltonians. David 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 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. ==In popular culture==

The butterfly effect has appeared across media such as literature (for instance, A Sound of Thunder), films and television (such as The Simpsons), video games (such as Life Is Strange), webcomics (such as Homestuck), musical references (such as "Butterfly Effect" by Travis Scott), AI-driven expansive language models, and more. ==See also==