Butterfly effect

Last updated

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.

Contents

The term butterfly effect is closely associated with the work of Edward Lorenz. It 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 that 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 unrounded initial condition data. A very small change in initial conditions had created a significantly different outcome. [1]

The idea that small causes may have large effects in weather was earlier recognized by French mathematician and engineer Henri Poincaré. 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. [2]

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 evidenced by the case of the three-body problem by Henri Poincaré in 1890. [3] He later proposed that such phenomena could be common, for example, in meteorology. [4]

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. [3]

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. [5]

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. [6]

Lorenz wrote:

In 1963, Lorenz published a theoretical study of this effect in a highly cited, seminal paper called Deterministic Nonperiodic Flow [8] [9] (the calculations were performed on a Royal McBee LGP-30 computer). [10] [11] Elsewhere he stated:

One meteorologist remarked that if the theory were correct, one flap of a sea gull's wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls. [11]

Following suggestions from colleagues, in later speeches and papers Lorenz used the more poetic butterfly. According to Lorenz, when he failed to provide a title for a talk he was to present at the 139th meeting of the American Association for the Advancement of Science in 1972, Philip Merrilees concocted Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? as a title. [12] Although a butterfly flapping its wings has remained constant in the expression of this concept, the location of the butterfly, the consequences, and the location of the consequences have varied widely. [13]

The phrase refers to the idea that a butterfly's wings might create 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 inter-connected complex web; one set of conditions leads to a tornado while the other set of conditions doesn't. The flapping wing represents 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. [14]

Some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed. [15] David Orrell argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role. [16] [17] 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. [18]

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 [19] to work he published in 1969 [20] 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. [21] [22]

Illustration

The butterfly effect in the Lorenz attractor
time 0  t  30 (larger) z coordinate (larger)
These figures show two segments of the three-dimensional evolution of two trajectories (one in blue, and the other in yellow) for the same period of time in the Lorenz attractor starting at two initial points that differ by only 10−5 in the x-coordinate. Initially, the two trajectories seem coincident, as indicated by the small difference between the z coordinate of the blue and yellow trajectories, but for t > 23 the difference is as large as the value of the trajectory. The final position of the cones indicates that the two trajectories are no longer coincident at t = 30.
An animation of the Lorenz attractor shows the continuous evolution.

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 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.

If M is the state space for the map ${\displaystyle f^{t}}$, then ${\displaystyle f^{t}}$ displays sensitive dependence to initial conditions if for any x in M and any δ > 0, there are y in M, with distance d(. , .) such that ${\displaystyle 0 and such that

${\displaystyle d(f^{\tau }(x),f^{\tau }(y))>\mathrm {e} ^{a\tau }\,d(x,y)}$

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.

The simplest mathematical framework exhibiting sensitive dependence on initial conditions is provided by a particular parametrization of the logistic map:

${\displaystyle x_{n+1}=4x_{n}(1-x_{n}),\quad 0\leq x_{0}\leq 1,}$

which, unlike most chaotic maps, has a closed-form solution:

${\displaystyle x_{n}=\sin ^{2}(2^{n}\theta \pi )}$

where the initial condition parameter ${\displaystyle \theta }$ is given by ${\displaystyle \theta ={\tfrac {1}{\pi }}\sin ^{-1}(x_{0}^{1/2})}$. For rational ${\displaystyle \theta }$, after a finite number of iterations ${\displaystyle x_{n}}$ maps into a periodic sequence. But almost all ${\displaystyle \theta }$ are irrational, and, for irrational ${\displaystyle \theta }$, ${\displaystyle x_{n}}$ 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 ${\displaystyle x_{n}}$ folded within the range [0, 1].

In physical systems

In 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." [23]

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. [24] [25] Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments; [26] [27] however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by Martin Gutzwiller [28] and Delos and co-workers. [29] The random matrix theory and simulations with quantum computers prove that some versions of the butterfly effect in quantum mechanics do not exist. [30]

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 different Hamiltonians. They investigate the level of sensitivity of quantum systems to small changes in their given Hamiltonians. [31] 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". [32] 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. [33] [34] This quantum butterfly effect has been demonstrated experimentally. [35] Quantum and semiclassical treatments of system sensitivity to initial conditions are known as quantum chaos. [26] [33]

The journalist Peter Dizikes, writing in The Boston Globe in 2008, notes that popular culture likes the idea of the butterfly effect, but gets it wrong. Whereas Lorenz suggested correctly with his butterfly metaphor that predictability "is inherently limited", popular culture supposes that each event can be explained by finding the small reasons that caused it. Dizikes explains: "It speaks to our larger expectation that the world should be comprehensible – that everything happens for a reason, and that we can pinpoint all those reasons, however small they may be. But nature itself defies this expectation." [36]

Related Research Articles

Chaos theory is a branch of mathematics focusing on the study of chaos—states of dynamical systems whose apparently random states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state. A metaphor for this behavior is that a butterfly flapping its wings in China can cause a hurricane in Texas.

The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst. Mathematically, the logistic map is written

In physics and mathematics, in the area of dynamical systems, a double pendulum is a pendulum with another pendulum attached to its end, and is a simple physical system that exhibits rich dynamic behavior with a strong sensitivity to initial conditions. The motion of a double pendulum is governed by a set of coupled ordinary differential equations and is chaotic.

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

Predictability is the degree to which a correct prediction or forecast of a system's state can be made either qualitatively or quantitatively.

An airfoil or aerofoil is the cross-sectional shape of a wing, blade, or sail.

In the study of dynamical systems, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes, but it does not preserve the geometric shape of structures in phase space.

The adiabatic theorem is a concept in quantum mechanics. Its original form, due to Max Born and Vladimir Fock (1928), was stated as follows:

Edward Norton Lorenz was an American mathematician and meteorologist who established the theoretical basis of weather and climate predictability, as well as the basis for computer-aided atmospheric physics and meteorology. He is best known as the founder of modern chaos theory, a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.

Recurrence quantification analysis (RQA) is a method of nonlinear data analysis for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a dynamical system presented by its phase space trajectory.

Ensemble forecasting is a method used in or within numerical weather prediction. Instead of making a single forecast of the most likely weather, a set of forecasts is produced. This set of forecasts aims to give an indication of the range of possible future states of the atmosphere. Ensemble forecasting is a form of Monte Carlo analysis. The multiple simulations are conducted to account for the two usual sources of uncertainty in forecast models: (1) the errors introduced by the use of imperfect initial conditions, amplified by the chaotic nature of the evolution equations of the atmosphere, which is often referred to as sensitive dependence on initial conditions; and (2) errors introduced because of imperfections in the model formulation, such as the approximate mathematical methods to solve the equations. Ideally, the verified future atmospheric state should fall within the predicted ensemble spread, and the amount of spread should be related to the uncertainty (error) of the forecast. In general, this approach can be used to make probabilistic forecasts of any dynamical system, and not just for weather prediction.

Quantum metrology is the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems, particularly exploiting quantum entanglement and quantum squeezing. This field promises to develop measurement techniques that give better precision than the same measurement performed in a classical framework. Together with quantum hypothesis testing, it represents an important theoretical model at the basis of quantum sensing.

The Lorenz system is a system of ordinary differential equations first studied by Edward Lorenz and Ellen Fetter. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. In popular media the 'butterfly effect' stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions, our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.

In physics and mathematics, in the area of dynamical systems, an elastic pendulum is a physical system where a piece of mass is connected to a spring so that the resulting motion contains elements of both a simple pendulum and a one-dimensional spring-mass system. The system exhibits chaotic behaviour and is sensitive to initial conditions. The motion of an elastic pendulum is governed by a set of coupled ordinary differential equations.

The butterfly effect is the phenomenon in chaos theory whereby a minor change in circumstances can cause a large change in outcome. The butterfly metaphor was created by Edward Norton Lorenz to emphasize the inherent unpredictable results of small changes in the initial conditions of certain physical systems. The concept was taken up by popular culture, and interpreted to mean that each event could be explained by some small cause, or that small events have a rippling effect that causes much larger events to take place.

The spin stiffness or spin rigidity or helicity modulus or the "superfluid density" is a constant which represents the change in the ground state energy of a spin system as a result of introducing a slow in plane twist of the spins. The importance of this constant is in its use as an indicator of quantum phase transitions—specifically in models with metal-insulator transitions such as Mott insulators. It is also related to other topological invariants such as the Berry phase and Chern numbers as in the Quantum hall effect.

Linear Optical Quantum Computing or Linear Optics Quantum Computation (LOQC) is a paradigm of quantum computation, allowing universal quantum computation. LOQC uses photons as information carriers, mainly uses linear optical elements, or optical instruments to process quantum information, and uses photon detectors and quantum memories to detect and store quantum information.

The KLM scheme or KLM protocol is an implementation of linear optical quantum computing (LOQC), developed in 2000 by Knill, Laflamme and Milburn. This protocol makes it possible to create universal quantum computers solely with linear optical tools. The KLM protocol uses linear optical elements, single photon sources and photon detectors as resources to construct a quantum computation scheme involving only ancilla resources, quantum teleportations and error corrections.

Quantum counting algorithm is a quantum algorithm for efficiently counting the number of solutions for a given search problem. The algorithm is based on the quantum phase estimation algorithm and on Grover's search algorithm.

Supersymmetric theory of stochastic dynamics or stochastics (STS) is an exact theory of stochastic (partial) differential equations (SDEs), the class of mathematical models with the widest applicability covering, in particular, all continuous time dynamical systems, with and without noise. The main utility of the theory from the physical point of view is a rigorous theoretical explanation of the ubiquitous spontaneous long-range dynamical behavior that manifests itself across disciplines via such phenomena as 1/f, flicker, and crackling noises and the power-law statistics, or Zipf's law, of instantonic processes like earthquakes and neuroavalanches. From the mathematical point of view, STS is interesting because it bridges the two major parts of mathematical physics – the dynamical systems theory and topological field theories. Besides these and related disciplines such as algebraic topology and supersymmetric field theories, STS is also connected with the traditional theory of stochastic differential equations and the theory of pseudo-Hermitian operators.

References

1. Lorenz, Edward N. (March 1963). "Deterministic Nonperiodic Flow". Journal of the Atmospheric Sciences. 20 (2): 130–141. Bibcode:1963JAtS...20..130L. doi:.
2. "Butterfly effect - Scholarpedia". www.scholarpedia.org. Archived from the original on 2016-01-02. Retrieved 2016-01-02.
3. Steves, Bonnie; Maciejewski, AJ (September 2001). The Restless Universe Applications of Gravitational N-Body Dynamics to Planetary Stellar and Galactic Systems. USA: CRC Press. ISBN   0750308222 . Retrieved January 6, 2014.
4. Flam, Faye (2012-06-15). "The Physics of Ray Bradbury's "A Sound of Thunder"". The Philadelphia Inquirer . Archived from the original on 2015-09-24. Retrieved 2015-09-02.
5. Gleick, James (1987). Chaos: Making a New Science. Viking. p. 16. ISBN   0-8133-4085-3.
6. Motter, Adilson E.; Campbell, David K. (2013). "Chaos at fifty". Physics Today. 66 (5): 27–33. arXiv:. Bibcode:2013PhT....66e..27M. doi:10.1063/PT.3.1977.
7. Lorenz, Edward N. (March 1963). "Deterministic Nonperiodic Flow". Journal of the Atmospheric Sciences. 20 (2): 130–141. Bibcode:1963JAtS...20..130L. doi:. ISSN   1520-0469.
8. "Part19". Cs.ualberta.ca. 1960-11-22. Archived from the original on 2009-07-17. Retrieved 2014-06-08.
9. Lorenz, Edward N. (1963). "The Predictability of Hydrodynamic Flow" (PDF). Transactions of the New York Academy of Sciences. 25 (4): 409–432. doi:10.1111/j.2164-0947.1963.tb01464.x. Archived (PDF) from the original on 10 October 2014. Retrieved 1 September 2014.
10. Lorenz: "Predictability", AAAS 139th meeting, 1972 Archived 2013-06-12 at the Wayback Machine Retrieved May 22, 2015
11. "The Butterfly Effects: Variations on a Meme". AP42 ...and everything. Archived from the original on 11 November 2011. Retrieved 3 August 2011.
12. Woods, Austin (2005). . New York: Springer. p.  118. ISBN   978-0387269283.
13. Orrell, David; Smith, Leonard; Barkmeijer, Jan; Palmer, Tim (2001). "Model error in weather forecasting". Nonlinear Processes in Geophysics. 9 (6): 357–371. Bibcode:2001NPGeo...8..357O. doi:.
14. Orrell, David (2002). "Role of the metric in forecast error growth: How chaotic is the weather?". Tellus. 54A (4): 350–362. Bibcode:2002TellA..54..350O. doi:.
15. Orrell, David (2012). Truth or Beauty: Science and the Quest for Order. New Haven: Yale University Press. p. 208. ISBN   978-0300186611.
16. Wolfram, Stephen (2002). . Wolfram Media. p.  998. ISBN   978-1579550080.
17. Lorenz: "Predictability", AAAS 139th meeting, 1972 Archived 2013-06-12 at the Wayback Machine Retrieved May 22, 2015
18. Lorenz, Edward N. (June 1969). "The predictability of a flow which possesses many scales of motion". Tellus. XXI (3): 289–297. Bibcode:1969TellA..21..289L. doi:10.1111/j.2153-3490.1969.tb00444.x.
19. Tim, Palmer (19 May 2017). "The Butterfly Effect - What Does It Really Signify?". Oxford U. Dept. of Mathematics Youtube Channel. Retrieved 13 February 2019.
20. Emanuel, Kerry (26 March 2018). "Edward N. Lorenz and the End of the Cartesian Universe". MIT Department of Earth, Atmospheric, and Planetary Sciences Youtube channel. Retrieved 13 February 2019.
21. "Chaos and Climate". RealClimate. Archived from the original on 2014-07-02. Retrieved 2014-06-08.
22. Heller, E. J.; Tomsovic, S. (July 1993). "Postmodern Quantum Mechanics". Physics Today . 46 (7): 38–46. Bibcode:1993PhT....46g..38H. doi:10.1063/1.881358.
23. Gutzwiller, Martin C. (1990). Chaos in Classical and Quantum Mechanics. New York: Springer-Verlag. ISBN   0-387-97173-4.
24. Rudnick, Ze'ev (January 2008). "What is...Quantum Chaos" (PDF). Notices of the American Mathematical Society. Archived (PDF) from the original on 2009-10-02.
25. Berry, Michael (1989). "Quantum chaology, not quantum chaos". Physica Scripta. 40 (3): 335–336. Bibcode:1989PhyS...40..335B. doi:10.1088/0031-8949/40/3/013.
26. Gutzwiller, Martin C. (1971). "Periodic Orbits and Classical Quantization Conditions". Journal of Mathematical Physics . 12 (3): 343. Bibcode:1971JMP....12..343G. doi:10.1063/1.1665596.
27. Gao, J. & Delos, J. B. (1992). "Closed-orbit theory of oscillations in atomic photoabsorption cross sections in a strong electric field. II. Derivation of formulas". Physical Review A . 46 (3): 1455–1467. Bibcode:1992PhRvA..46.1455G. doi:10.1103/PhysRevA.46.1455. PMID   9908268.
28. Yan, Bin; Sinitsyn, Nikolai A. (2020). "Recovery of Damaged Information and the Out-of-Time-Ordered Correlators". Physical Review Letters. 125: 040605. arXiv:. doi:10.1103/PhysRevLett.125.040605.
29. Karkuszewski, Zbyszek P.; Jarzynski, Christopher; Zurek, Wojciech H. (2002). "Quantum Chaotic Environments, the Butterfly Effect, and Decoherence". Physical Review Letters . 89 (17): 170405. arXiv:. Bibcode:2002PhRvL..89q0405K. doi:10.1103/PhysRevLett.89.170405. PMID   12398653.
30. Poulin, David; Blume-Kohout, Robin; Laflamme, Raymond & Ollivier, Harold (2004). "Exponential Speedup with a Single Bit of Quantum Information: Measuring the Average Fidelity Decay". Physical Review Letters. 92 (17): 177906. arXiv:. Bibcode:2004PhRvL..92q7906P. doi:10.1103/PhysRevLett.92.177906. PMID   15169196.
31. Poulin, David. "A Rough Guide to Quantum Chaos" (PDF). Archived from the original (PDF) on 2010-11-04.
32. Peres, A. (1995). Quantum Theory: Concepts and Methods . Dordrecht: Kluwer Academic.
33. Lee, Jae-Seung & Khitrin, A. K. (2004). "Quantum amplifier: Measurement with entangled spins". Journal of Chemical Physics . 121 (9): 3949–51. Bibcode:2004JChPh.121.3949L. doi:10.1063/1.1788661. PMID   15332940.
34. Dizikes, Petyer (8 June 2008). "The meaning of the butterfly". The Boston Globe. Archived from the original on 18 April 2016. Retrieved 8 June 2016.