Editing Chaos theory (section)

==History==
[[File:Barnsley fern plotted with VisSim.PNG|thumb|upright|[[Barnsley fern]] created using the [[chaos game]]. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an [[iterated function system]] (IFS).]]

[[James Clerk Maxwell]] first emphasized the "[[butterfly effect]]", and is seen as being one of the earliest to discuss chaos theory, with work in the 1860s and 1870s.<ref>{{Cite journal |last1=Hunt |first1=Brian R. |last2=Yorke |first2=James A. |date=1993 |title=Maxwell on Chaos |url=https://yorke.umd.edu/Yorke_papers_most_cited_and_post2000/1993_04_Hunt_%20Nonlin-Science-Today%20_Maxwell%20on%20Chaos.PDF |journal=Nonlinear Science Today |volume=3 |issue=1}}</ref><ref>{{Cite web |last=Everitt |first=Francis |date=2006-12-01 |title=James Clerk Maxwell: a force for physics |url=https://physicsworld.com/a/james-clerk-maxwell-a-force-for-physics/ |access-date=2023-11-03 |website=Physics World |language=en-GB}}</ref><ref>{{Cite journal |last1=Gardini |first1=Laura |last2=Grebogi |first2=Celso |last3=Lenci |first3=Stefano |date=2020-10-01 |title=Chaos theory and applications: a retrospective on lessons learned and missed or new opportunities |journal=Nonlinear Dynamics |language=en |volume=102 |issue=2 |pages=643–644 |doi=10.1007/s11071-020-05903-0 |s2cid=225246631 |issn=1573-269X|doi-access=free |bibcode=2020NonDy.102..643G |hdl=2164/17003 |hdl-access=free }}</ref> An early proponent of chaos theory was [[Henri Poincaré]]. In the 1880s, while studying the [[three-body problem]], he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point.<ref>{{cite journal |author=Poincaré, Jules Henri |title=Sur le problème des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt |journal=Acta Mathematica |volume=13 |issue=1–2 |pages=1–270 |year=1890 |doi=10.1007/BF02392506 |doi-access=free }}</ref><ref>{{Cite book|title=The three-body problem and the equations of dynamics : Poincaré's foundational work on dynamical systems theory|last=Poincaré|first=J. Henri|publisher=Springer International Publishing|others=Popp, Bruce D. (Translator)|year=2017|isbn=9783319528984|location=Cham, Switzerland|oclc=987302273}}</ref><ref>{{cite book |author1=Diacu, Florin |author2=Holmes, Philip |title=Celestial Encounters: The Origins of Chaos and Stability |publisher=[[Princeton University Press]] |year=1996 }}</ref> In 1898, [[Jacques Hadamard]] published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature, called "[[Hadamard's billiards]]".<ref>{{cite journal|first = Jacques|last = Hadamard|year = 1898|title = Les surfaces à courbures opposées et leurs lignes géodesiques|journal = Journal de Mathématiques Pures et Appliquées|volume = 4|pages = 27–73}}</ref> Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one another, with a positive [[Lyapunov exponent]].

Chaos theory began in the field of [[ergodic theory]]. Later studies, also on the topic of nonlinear [[differential equations]], were carried out by [[George David Birkhoff]],<ref>George D. Birkhoff, ''Dynamical Systems,'' vol.&nbsp;9 of the American Mathematical Society Colloquium Publications (Providence, Rhode Island: American Mathematical Society, 1927)</ref> [[Andrey Nikolaevich Kolmogorov]],<ref>{{cite journal| last=Kolmogorov | first=Andrey Nikolaevich | author-link=Andrey Nikolaevich Kolmogorov | year=1941 | title=Local structure of turbulence in an incompressible fluid for very large Reynolds numbers | journal=[[Doklady Akademii Nauk SSSR]] | volume=30 | issue=4 | pages=301–5 |bibcode = 1941DoSSR..30..301K | title-link=turbulence }} Reprinted in: {{cite journal |journal=Proceedings of the Royal Society A |volume=434 |pages=9–13 |year=1991 |doi=10.1098/rspa.1991.0075 |title=The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers |last1=Kolmogorov |first1=A. N. |s2cid=123612939 |issue=1890 |bibcode=1991RSPSA.434....9K}}</ref><ref>{{cite journal| last=Kolmogorov | first=A. N. | year=1941 | title=On degeneration of isotropic turbulence in an incompressible viscous liquid | journal=Doklady Akademii Nauk SSSR | volume=31 | issue=6 | pages=538–540}} Reprinted in: {{cite journal |journal=Proceedings of the Royal Society A |volume=434 |pages=15–17 |year=1991 |doi=10.1098/rspa.1991.0076 |title=Dissipation of Energy in the Locally Isotropic Turbulence |last1=Kolmogorov |first1=A. N. |s2cid=122060992 |issue=1890 |bibcode=1991RSPSA.434...15K}}</ref><ref>{{cite book| last=Kolmogorov | first=A. N. | title=Stochastic Behavior in Classical and Quantum Hamiltonian Systems | chapter=Preservation of conditionally periodic movements with small change in the Hamilton function|pages=51–56| bibcode=1979LNP....93...51K| doi=10.1007/BFb0021737| series=Lecture Notes in Physics| date=1979 | volume=93 | isbn=978-3-540-09120-2}} Translation of ''Doklady Akademii Nauk SSSR'' (1954) 98: 527. See also [[Kolmogorov–Arnold–Moser theorem]]</ref> [[Mary Lucy Cartwright]] and [[John Edensor Littlewood]],<ref>{{cite journal |last1=Cartwright |first1=Mary L. |last2=Littlewood |first2=John E. |title=On non-linear differential equations of the second order, I: The equation ''y''" + ''k''(1−''y''<sup>2</sup>)''y<nowiki>'</nowiki>'' + ''y'' = ''b''λkcos(λ''t'' + ''a''), ''k'' large |journal=Journal of the London Mathematical Society |volume=20 |pages=180–9 |year=1945 |doi=10.1112/jlms/s1-20.3.180 |issue=3 }} See also: [[Van der Pol oscillator]]</ref> and [[Stephen Smale]].<ref>{{cite journal |author=Smale, Stephen |title=Morse inequalities for a dynamical system |journal=Bulletin of the American Mathematical Society |volume=66 |pages=43–49 |date=January 1960 |doi=10.1090/S0002-9904-1960-10386-2 |doi-access=free }}</ref> Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.

Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that [[linear theory]], the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the [[logistic map]]. What had been attributed to measure imprecision and simple "[[Pseudorandom noise|noise]]" was considered by chaos theorists as a full component of the studied systems. In 1959 [[Boris Chirikov|Boris Valerianovich Chirikov]] proposed a criterion for the emergence of classical chaos in Hamiltonian systems ([[Chirikov criterion]]). He applied this criterion to explain some experimental results on [[plasma confinement]] in open mirror traps.<ref>{{Cite journal |last=Chirikov |first=Boris |title=РЕЗОНАНСНЫЕ ПРОЦЕССЫ В МАГНИТНЫХ ЛОВУШКАХ |url=https://www.quantware.ups-tlse.fr/chirikov/refs/chi1959.pdf |journal=Атомная энергия |volume=6}}</ref><ref>{{Cite journal |last=Chirikov |first=B. V. |date=1960-12-01 |title=Resonance processes in magnetic traps |url=https://doi.org/10.1007/BF01483352 |journal=The Soviet Journal of Atomic Energy |language=en |volume=6 |issue=6 |pages=464–470 |doi=10.1007/BF01483352 |s2cid=59483478 |issn=1573-8205}}</ref> This is regarded as the very first physical theory of chaos, which succeeded in explaining a concrete experiment. And Boris Chirikov himself is considered as a pioneer in classical and quantum chaos.<ref>{{Cite journal |last1=Jean |first1=Bellissard |author-link=Jean Bellissard |last2=Dima |first2=Shepelyansky |date=27 February 1998 |title=Boris Chirikov, a pioneer in classical and quantum chaos |url=https://www.quantware.ups-tlse.fr/chirikov/aboutchirikov/poincare1998.pdf |journal=[[Annales Henri Poincaré]] |volume=68 |issue=4 |page=379}}</ref><ref>{{Cite journal |last1=Bellissard |first1=J. |last2=Bohigas |first2=O. |last3=Casati |first3=G. |last4=Shepelyansky |first4=D.L. |date=1 July 1999 |title=A pioneer of chaos |url=http://dx.doi.org/10.1016/s0167-2789(99)90007-6 |journal=Physica D: Nonlinear Phenomena |volume=131 |issue=1–4 |pages=viii–xv |doi=10.1016/s0167-2789(99)90007-6 |bibcode=1999PhyD..131D...8B |s2cid=119107150 |issn=0167-2789}}</ref><ref>{{Cite book |first=Dima |last=Shepelyansky |url=http://worldcat.org/oclc/859751750 |title=Chaos at Fifty Four in 2013 |oclc=859751750}}</ref>

The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated [[iteration]] of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems. As a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers and noticed, on November 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970.<ref>{{harvnb|Abraham|Ueda|2000|loc=See Chapters 3 and 4}}</ref><ref>{{harvnb|Sprott|2003|p=[https://books.google.com/books?id=SEDjdjPZ158C&pg=PA89 89]}}</ref>

[[File:Airplane vortex edit.jpg|thumb|left|[[Turbulence]] in the [[Wingtip vortices|tip vortex]] from an [[airplane]] wing. Studies of the critical point beyond which a system creates turbulence were important for chaos theory, analyzed for example by the [[Soviet physicists|Soviet physicist]] [[Lev Landau]], who developed the [[Landau-Hopf theory of turbulence]]. [[David Ruelle]] and [[Floris Takens]] later predicted, against Landau, that [[fluid turbulence]] could develop through a [[strange attractor]], a main concept of chaos theory.]]

[[Edward Lorenz]] was an early pioneer of the theory. His interest in chaos came about accidentally through his work on [[meteorology|weather prediction]] in 1961.<ref>Lorenz, E.N. The statistical prediction of solutions of dynamic equations. In Proceedings of the International Symposium on Numerical Weather Prediction, Tokyo, Japan, 7–13 November 1962; pp. 629–635. {{Citation |title=Numerical solutions to the equations |date=2010-12-02 |work=Numerical Weather and Climate Prediction |pages=17–118 |url=https://doi.org/10.1017/cbo9780511763243.004 |access-date=2025-03-05 |publisher=Cambridge University Press|doi=10.1017/cbo9780511763243.004 |isbn=978-0-521-51389-0 }}</ref><ref name=Lorenz1961>{{cite journal |author=Lorenz, Edward N. |title=Deterministic non-periodic flow |journal=Journal of the Atmospheric Sciences |volume=20 |pages=130–141 |year=1963 |doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |issue=2 |bibcode=1963JAtS...20..130L|doi-access=free }}</ref> Lorenz and his collaborator [[Ellen Fetter]] and [[Margaret Hamilton (software engineer)|Margaret Hamilton]]<ref>{{Cite web |last=Sokol |first=Joshua |date=May 20, 2019 |title=The Hidden Heroines of Chaos |url=https://www.quantamagazine.org/the-hidden-heroines-of-chaos-20190520/ |access-date=2022-11-09 |website=Quanta Magazine}}</ref> were using a simple digital computer, a [[Royal McBee]] [[LGP-30]], to run weather simulations. They wanted to see a sequence of data again, and to save time they started the simulation in the middle of its course. They did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To their surprise, the weather the machine began to predict was completely different from the previous calculation. They tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome.<ref>{{cite book|title=Chaos: Making a New Science |last=Gleick |first=James |year=1987 |publisher=Cardinal |location=London|page=17|isbn=978-0-434-29554-8|title-link=Chaos: Making a New Science }}</ref> Lorenz's discovery, which gave its name to [[Lorenz attractor]]s, showed that even detailed atmospheric modeling cannot, in general, make precise long-term weather predictions.

In 1963, [[Benoit Mandelbrot]], studying [[information theory]], discovered that noise in many phenomena (including [[stock market|stock prices]] and [[telephone]] circuits) was patterned like a [[Cantor set]], a set of points with infinite roughness and detail.<ref>{{cite journal |author1=Berger J.M. |author2=Mandelbrot B. | year = 1963 | title = A new model for error clustering in telephone circuits | journal = IBM Journal of Research and Development | volume = 7 |issue=3 | pages = 224–236 | doi=10.1147/rd.73.0224}}</ref> Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards).<ref>{{cite book |author=Mandelbrot, B. |title=The Fractal Geometry of Nature |publisher=Freeman |location=New York |year=1977 |page=248 }}</ref><ref>See also: {{cite book |last1=Mandelbrot |first1=Benoît B. |last2=Hudson |first2=Richard L. |title=The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward |url=https://archive.org/details/misbehaviorofmar00beno |url-access=registration |publisher=Basic Books |location=New York |year=2004 |page=[https://archive.org/details/misbehaviorofmar00beno/page/201 201] |isbn=9780465043552 }}</ref> In 1967, he published "[[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|How long is the coast of Britain? Statistical self-similarity and fractional dimension]]", showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an [[infinitesimal]]ly small measuring device.<ref>{{cite journal |last=Mandelbrot |first=Benoît |title=How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension |journal=Science |volume=156 |issue=3775 |pages=636–8 |date=5 May 1967 |doi=10.1126/science.156.3775.636 |pmid=17837158 |bibcode=1967Sci...156..636M |s2cid=15662830 |url=http://ena.lp.edu.ua:8080/handle/ntb/52473 |access-date=31 January 2022 |archive-date=19 October 2021 |archive-url=https://web.archive.org/web/20211019193011/http://ena.lp.edu.ua:8080/handle/ntb/52473 |url-status=dead }}</ref> Arguing that a ball of twine appears as a point when viewed from far away (0-dimensional), a ball when viewed from fairly near (3-dimensional), or a curved strand (1-dimensional), he argued that the dimensions of an object are relative to the observer and may be fractional. An object whose irregularity is constant over different scales ("self-similarity") is a [[fractal]] (examples include the [[Menger sponge]], the [[Sierpiński gasket]], and the [[Koch curve]] or ''snowflake'', which is infinitely long yet encloses a finite space and has a [[fractal dimension]] of circa 1.2619). In 1982, Mandelbrot published ''[[The Fractal Geometry of Nature]]'', which became a classic of chaos theory.<ref>{{cite book|author=Mandelbrot, B.|title=The Fractal Geometry of Nature|date=1982|place=New York|publisher=Macmillan|isbn=978-0716711865|url=https://archive.org/details/fractalgeometryo00beno}}</ref>

In December 1977, the [[New York Academy of Sciences]] organized the first symposium on chaos, attended by David Ruelle, [[Robert May, Baron May of Oxford|Robert May]], [[James A. Yorke]] (coiner of the term "chaos" as used in mathematics), [[Robert Shaw (physicist)|Robert Shaw]], and the meteorologist Edward Lorenz. The following year Pierre Coullet and Charles Tresser published "Itérations d'endomorphismes et groupe de renormalisation", and [[Mitchell Feigenbaum]]'s article "Quantitative Universality for a Class of Nonlinear Transformations" finally appeared in a journal, after 3 years of referee rejections.<ref name="Feigenbaum 25–52">{{cite journal |first=Mitchell |last=Feigenbaum |title=Quantitative universality for a class of nonlinear transformations |journal=Journal of Statistical Physics |volume=19 |issue=1 |pages=25–52 |date=July 1978 |doi=10.1007/BF01020332 |bibcode=1978JSP....19...25F|citeseerx=10.1.1.418.9339 |s2cid=124498882 }}</ref><ref>Coullet, Pierre, and Charles Tresser. "Iterations d'endomorphismes et groupe de renormalisation." Le Journal de Physique Colloques 39.C5 (1978): C5-25</ref> Thus Feigenbaum (1975) and Coullet & Tresser (1978) discovered the [[Universality (dynamical systems)|universality]] in chaos, permitting the application of chaos theory to many different phenomena.

In 1979, [[Albert J. Libchaber]], during a symposium organized in Aspen by [[Pierre Hohenberg]], presented his experimental observation of the [[Bifurcation theory|bifurcation]] cascade that leads to chaos and turbulence in [[Rayleigh–Bénard convection]] systems. He was awarded the [[Wolf Prize in Physics]] in 1986 along with [[Mitchell J. Feigenbaum]] for their inspiring achievements.<ref>{{cite web|url = http://www.wolffund.org.il/cat.asp?id=25&cat_title=PHYSICS|title = The Wolf Prize in Physics in 1986.|access-date = 2008-01-17|archive-date = 2024-05-25|archive-url = https://archive.today/20240525172450/https://www.webcitation.org/65DR3JlSI?url=http://www.wolffund.org.il/cat.asp%3Fid=25|url-status = dead}}</ref>

In 1986, the New York Academy of Sciences co-organized with the [[National Institute of Mental Health]] and the [[Office of Naval Research]] the first important conference on chaos in biology and medicine. There, [[Bernardo Huberman]] presented a mathematical model of the [[Eye movement|eye tracking]] dysfunction among people with [[schizophrenia]].<ref>{{cite journal |author-link=Bernardo Huberman |author=Huberman, B.A. |title=A Model for Dysfunctions in Smooth Pursuit Eye Movement |journal=Annals of the New York Academy of Sciences |volume=504 Perspectives in Biological Dynamics and Theoretical Medicine |issue=1 |pages=260–273 |date=July 1987 |doi=10.1111/j.1749-6632.1987.tb48737.x |pmid=3477120 |bibcode = 1987NYASA.504..260H |s2cid=42733652 }}</ref> This led to a renewal of [[physiology]] in the 1980s through the application of chaos theory, for example, in the study of pathological [[cardiac cycle]]s.

In 1987, [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] published a paper in ''[[Physical Review Letters]]''<ref>{{cite journal |author1=Bak, Per |author2=Tang, Chao |author3=Wiesenfeld, Kurt |title=Self-organized criticality: An explanation of the 1/f noise |journal=Physical Review Letters |volume=59 |issue=4 |pages=381–4 |date=27 July 1987 |doi=10.1103/PhysRevLett.59.381 |pmid=10035754 |bibcode=1987PhRvL..59..381B|s2cid=7674321 }} However, the conclusions of this article have been subject to dispute. {{cite web|url=http://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html |title=? |url-status=dead |archive-url=https://web.archive.org/web/20071214033929/https://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html |archive-date=2007-12-14 }}. See especially: {{cite journal |author1=Laurson, Lasse |author2=Alava, Mikko J. |author3=Zapperi, Stefano |title=Letter: Power spectra of self-organized critical sand piles |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=0511 |id=L001 |date=15 September 2005 }}</ref> describing for the first time [[self-organized criticality]] (SOC), considered one of the mechanisms by which [[complexity]] arises in nature.

Alongside largely lab-based approaches such as the [[Bak–Tang–Wiesenfeld sandpile]], many other investigations have focused on large-scale natural or social systems that are known (or suspected) to display [[scale invariance|scale-invariant]] behavior. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including [[earthquake]]s, (which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the [[Gutenberg–Richter law]] describing the statistical distribution of earthquake sizes, and the [[Aftershock|Omori law]]<ref>{{cite journal |author=Omori, F. |title=On the aftershocks of earthquakes |journal=Journal of the College of Science, Imperial University of Tokyo |volume=7 |pages=111–200 |year=1894 }}</ref> describing the frequency of aftershocks), [[solar flare]]s, fluctuations in economic systems such as [[financial market]]s (references to SOC are common in [[econophysics]]), landscape formation, [[forest fire]]s, [[landslide]]s, [[epidemic]]s, and [[biological evolution]] (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "[[punctuated equilibrium|punctuated equilibria]]" put forward by [[Niles Eldredge]] and [[Stephen Jay Gould]]). Given the implications of a scale-free distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of [[war]]s. These investigations of SOC have included both attempts at modelling (either developing new models or adapting existing ones to the specifics of a given natural system), and extensive data analysis to determine the existence and/or characteristics of natural scaling laws.
 
Also in 1987 [[James Gleick]] published ''[[Chaos: Making a New Science]]'', which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public.<ref name=":8">{{cite book|last=Gleick|first=James|title=Chaos: Making a New Science|date=August 26, 2008|publisher=Penguin Books|isbn=978-0143113454}}</ref> Initially the domain of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of [[nonlinear system]]s analysis. Alluding to [[Thomas Kuhn]]'s concept of a [[paradigm shift]] exposed in ''[[The Structure of Scientific Revolutions]]'' (1962), many "chaologists" (as some described themselves) claimed that this new theory was an example of such a shift, a thesis upheld by Gleick.

The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory remains an active area of research,<ref>{{cite journal | last1 = Motter | first1 = A. E. | last2 = Campbell | first2 = D. K. | year = 2013 | title = Chaos at fifty | url = http://www.physicstoday.org/resource/1/phtoad/v66/i5/p27_s1?bypassSSO=1 | journal = Phys. Today | volume = 66 | issue = 5| pages = 27–33 | doi=10.1063/pt.3.1977|arxiv = 1306.5777 |bibcode = 2013PhT....66e..27M | s2cid = 54005470 }}</ref> involving many different disciplines such as [[mathematics]], [[topology]], [[physics]],<ref>{{cite journal|last1=Hubler|first1=A.|last2=Foster|first2=G.|last3=Phelps|first3=K.|title=Managing chaos: Thinking out of the box|journal=Complexity|volume=12|issue=3|pages=10|date=2007|doi=10.1002/cplx.20159|bibcode = 2007Cmplx..12c..10H }}</ref> [[social systems]],<ref>{{Cite book|title=Chaos Theory in the Social Sciences: Foundations and Applications|date=1996|publisher=University of Michigan Press|isbn=9780472106387|editor-last=Kiel|editor-first=L.|location=Ann Arbor, MI|language=en|doi=10.3998/mpub.14623|editor-last2=Elliott|editor-first2=Euel|hdl = 2027/fulcrum.d504rm03n}}</ref> [[population model]]ing, [[biology]], [[meteorology]], [[astrophysics]], [[information theory]], [[computational neuroscience]], [[pandemic]] [[crisis management]],<ref name="CT-REF-20"/><ref name="CT-REF-21"/> etc.