Chaos论文

混沌学 Chaos theoryIn mathematics, chaos theory describes the behavior of certain dynamical systems – that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the butterfly effect) As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears to be This happens even though these systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements This behavior is known as deterministic chaos, or simply Chaotic behaviour is also observed in natural systems, such as the This may be explained by a chaos-theoretical analysis of a mathematical model of such a system, embodying the laws of physics that are relevant for the natural OverviewChaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, oscillating chemical reactions, fluid dynamics, and mechanical and magneto-mechanical Observations of chaotic behaviour in nature include the dynamics of satellites in the solar system, the time evolution of the magnetic field of celestial bodies, population growth in ecology, the dynamics of the action potentials in neurons, and molecular Everyday examples of chaotic systems include weather and [1] There is some controversy over the existence of chaotic dynamics in the plate tectonics and in [2][3][4]Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete A related field of physics called quantum chaos theory studies systems that follow the laws of quantum Recently, another field, called relativistic chaos,[5] has emerged to describe systems that follow the laws of general As well as being orderly in the sense of being deterministic, chaotic systems usually have well defined [citation needed] For example, the Lorenz system pictured is chaotic, but has a clearly defined Bounded chaos is a useful term for describing models of HistoryThe first discoverer of chaos was Henri Poincaré In 1890, while studying the three-body problem, he found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed [6] In 1898 Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative [7] In the system studied, "Hadamard's billiards," Hadamard was able to show that all trajectories are unstable in that all particle trajectories diverge exponentially from one another, with a positive Lyapunov Much of the earlier theory was developed almost entirely by mathematicians, under the name of ergodic Later studies, also on the topic of nonlinear differential equations, were carried out by GD Birkhoff,[8] A N Kolmogorov,[9][10][11] ML Cartwright and JE Littlewood,[12] and Stephen S[13] Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and L 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 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 for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behaviour of certain experiments like that of the logistic What had been beforehand excluded as measure imprecision and simple "noise" was considered by chaos theories as a full component of the studied The main catalyst for the development of chaos theory was the electronic Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these One of the earliest electronic digital computers, ENIAC, was used to run simple weather forecasting An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in [14] Lorenz was using a simple digital computer, a Royal McBee LGP-30, to run his weather He wanted to see a sequence of data again and to save time he started the simulation in the middle of its He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last To his surprise the weather that the machine began to predict was completely different from the weather calculated Lorenz tracked this down to the computer The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 506127 was printed as This difference is tiny and the consensus at the time would have been that it should have had practically no However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term [15] Lorenz's discovery, which gave its name to Lorenz attractors, proved that meteorology could not reasonably predict weather beyond a weekly period (at most)The year before, Benoit Mandelbrot found recurring patterns at every scale in data on cotton [16] Beforehand, he had studied information theory and concluded noise was patterned like a Cantor set: on any scale the proportion of noise-containing periods to error-free periods was a constant – thus errors were inevitable and must be planned for by incorporating [17] Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur, , in a stock's prices after bad news, thus challenging normal distribution theory in statistics, aka Bell Curve) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards)[18][19] In 1967, he published "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 infinitesimally small measuring [20] Arguing that a ball of twine appears to be 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 An object whose irregularity is constant over different scales ("self-similarity") is a fractal (for example, the Koch curve or "snowflake", which is infinitely long yet encloses a finite space and has fractal dimension equal to circa 2619, the Menger sponge and the Sierpiński gasket) In 1975 Mandelbrot published The Fractal Geometry of Nature, which became a classic of chaos Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal Chaos was observed by a number of experimenters before it was recognized; , in 1927 by van der Pol[21] and in 1958 by RL I[22][23] However, Yoshisuke Ueda seems to have been the first experimenter to have identified a chaotic phenomenon as such by using an analog computer on November 27, The chaos exhibited by an analog computer is a real phenomenon, in contrast with those that digital computers calculate, which has a different kind of limit on Ueda's supervising professor, Hayashi, did not believe in chaos, and thus he prohibited Ueda from publishing his findings until [24]In December 1977 the New York Academy of Sciences organized the first symposium on Chaos, attended by David Ruelle, Robert May, James Yorke (coiner of the term "chaos" as used in mathematics), Robert Shaw (a physicist, part of the Eudaemons group with J Doyne Farmer and Norman Packard who tried to find a mathematical method to beat roulette, and then created with them the Dynamical Systems Collective in Santa Cruz), and the meteorologist Edward LThe following year, Mitchell Feigenbaum published the noted article "Quantitative Universality for a Class of Nonlinear Transformations", where he described logistic [25] Feigenbaum had applied fractal geometry to the study of natural forms such as Feigenbaum notably discovered the universality in chaos, permitting an application of chaos theory to many different In 1979, Albert J Libchaber, during a symposium organized in Aspen by Pierre Hohenberg, presented his experimental observation of the bifurcation cascade that leads to chaos and turbulence in convective Rayleigh–Benard He was awarded the Wolf Prize in Physics in 1986 along with Mitchell J Feigenbaum "for his brilliant experimental demonstration of the transition to turbulence and chaos in dynamical systems"[26]The New York Academy of Sciences then co-organized, in 1986, with the National Institute of Mental Health and the Office of Naval Research the first important conference on Chaos in biology and Bernardo Huberman thereby presented a mathematical model of the eye tracking disorder among [27] Chaos theory thereafter renewed physiology in the 1980s, for example in the study of pathological cardiac In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters[28] describing for the first time self-organized criticality (SOC), considered to be one of the mechanisms by which complexity arises in Alongside largely lab-based approaches such as the Bak–Tang–Wiesenfeld sandpile, many other investigations have centred around large-scale natural or social systems that are known (or suspected) to display scale-invariant 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: earthquakes (which, long before SOC was discovered, were known as a source of scale-invariant behaviour such as the Gutenberg–Richter law describing the statistical distribution of earthquake sizes, and the Omori law[29] describing the frequency of aftershocks); solar flares; fluctuations in economic systems such as financial markets (references to SOC are common in econophysics); landscape formation; forest fires; landslides; epidemics; and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "punctuated equilibria" put forward by Niles Eldredge and Stephen Jay Gould) Worryingly, 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 These "applied" 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 The same year, James Gleick published Chaos: Making a New Science, which became a best-seller and introduced general principles of chaos theory as well as its history to the broad At first the domains of work of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of nonlinear systems Alluding to Thomas Kuhn's concept of a paradigm shift exposed in The Structure of Scientific Revolutions (1962), many "chaologists" (as some self-nominated themselves) claimed that this new theory was an example of such as shift, a thesis upheld by J GThe availability of cheaper, more powerful computers broadens the applicability of chaos Currently, chaos theory continues to be a very active area of research, involving many different disciplines (mathematics, topology, physics, population biology, biology, meteorology, astrophysics, information theory, )[edit] Chaotic dynamicsFor a dynamical system to be classified as chaotic, it must have the following properties:[30]it must be sensitive to initial conditions, it must be topologically mixing, and its periodic orbits must be Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future Sensitivity to initial conditions is popularly known as the "butterfly effect", so called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, DC entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale Had the butterfly not flapped its wings, the trajectory of the system might have been vastly Sensitivity to initial conditions is often confused with chaos in popular It can also be a subtle property, since it depends on a choice of metric, or the notion of distance in the phase space of the For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by iterating the mapping on the real line that maps x to 2x) This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely However, it has extremely simple behaviour, as all points except 0 tend to If instead we use the bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle, the system no longer is sensitive to initial For this reason, in defining chaos, attention is normally restricted to systems with bounded metrics, or closed, bounded invariant subsets of unbounded Even for bounded systems, sensitivity to initial conditions is not identical with For example, consider the two-dimensional torus described by a pair of angles (x,y), each ranging between zero and 2π Define a mapping that takes any point (x,y) to (2x, y + a), where a is any number such that a/2π is Because of the doubling in the first coordinate, the mapping exhibits sensitive dependence on initial However, because of the irrational rotation in the second coordinate, there are no periodic orbits, and hence the mapping is not chaotic according to the definition Topologically mixing means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given Here, "mixing" is really meant to correspond to the standard intuition: the mixing of colored dyes or fluids is an example of a chaotic Linear systems are never chaotic; for a dynamical system to display chaotic behaviour it has to be Also, by the Poincaré–Bendixson theorem, a continuous dynamical system on the plane cannot be chaotic; among continuous systems only those whose phase space is non-planar (having dimension at least three, or with a non-Euclidean geometry) can exhibit chaotic However, a discrete dynamical system (such as the logistic map) can exhibit chaotic behaviour in a one-dimensional or two-dimensional phase [edit] AttractorsSome dynamical systems are chaotic everywhere (see Anosov diffeomorphisms) but in many cases chaotic behaviour is found only in a subset of phase The cases of most interest arise when the chaotic behaviour takes place on an attractor, since then a large set of initial conditions will lead to orbits that converge to this chaotic An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent Because of the topological transitivity condition, this is likely to produce a picture of the entire final For instance, in a system describing a pendulum, the phase space might be two-dimensional, consisting of information about position and One might plot the position of a pendulum against its A pendulum at rest will be plotted as a point, and one in periodic motion will be plotted as a simple closed When such a plot forms a closed curve, the curve is called an Our pendulum has an infinite number of such orbits, forming a pencil of nested ellipses about the [edit] Strange attractorsWhile most of the motion types mentioned above give rise to very simple attractors, such as points and circle-like curves called limit cycles, chaotic motion gives rise to what are known as strange attractors, attractors that can have great detail and For instance, a simple three-dimensional model of the Lorenz weather system gives rise to the famous Lorenz The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a Another such attractor is the Rössler map, which experiences period-two doubling route to chaos, like the logistic Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map) Other discrete dynamical systems have a repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points - Julia sets can be thought of as strange Both strange attractors and Julia sets typically have a fractal The Poincaré-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system if it has three or more However, no such restriction applies to discrete systems, which can exhibit strange attractors in two or even one dimensional The initial conditions of three or more bodies interacting through gravitational attraction (see the n-body problem) can be arranged to produce chaotic Minimum complexity of a chaotic systemSimple systems can also produce chaos without relying on differential An example is the logistic map, which is a difference equation (recurrence relation) that describes population growth over Another example is the Ricker model of population Even the evolution of simple discrete systems, such as cellular automata, can heavily depend on initial Stephen Wolfram has investigated a cellular automaton with this property, termed by him rule A minimal model for conservative (reversible) chaotic behavior is provided by Arnold's cat

混沌在数学分析上表现为迭代数列在初始值不同时其收敛性的不同,以及对不动点和周期的研究,可参阅谢惠民等编的<数学分析习题课讲义>中6和6两节。

Noether's theorem 很恶心阿，关键大家都明白还要硬严密推导。如果你擅长用微小量计算，建议选这个，然后导出微小平移和旋转的向量表达式。个人觉得可以把the works of Lagrange 和the works of Hamilton 结合起来解释某些具体问题。这里works不是著作是功，也就是著名的Lagrange 和 Hamilton 最典型的就是单摆了，dp/dt=-dH/dq,dq/dt=dH/[p,q]=可以从不同的角度解析，并给出p-qGraph,就是那个一圈一圈的，按E的不同分3种情况讨论运动。最后在深入，如果角度不可近似，那么就要用到chaos理论的分歧方程式，其实就是椭圆函数拉。应该可以把内容融会起来。

Noether's theorem the origin 开普勒定律-the origin Kepler's laws 拉格朗日的著作-the works of Lagrange 哈密顿的著作-the works of Hamilton 最小值原理-the least action principle numerical simulation of the Hohmann transfer orbit numerical simulation of a chaotic numerical simulation of motion near a Lagrange point 开普勒的三大定律 17世纪初期，正当伽利略使哥白尼学说声威大振之时，欧洲大地上传出了 一条特大新闻：德国天文学家约翰内斯。开普勒发现了行星运动的三大定律，使 哥白尼创立的日心说，从科学上向前前进了一步。 开普勒于1571年12月27日生于德国符腾堡的小城魏尔。幼年时，由 于家境贫寒，他一直靠奖学金上学。 后来，开普勒进人图宾根神学院后，在老师迈克尔的指导下，开始研究哥白 尼的天文学。1594年，开普勒成为奥地利格拉茨新教神学院的数学教师。 在这一时期，开普勒孜孜不倦地研究天文学的三个问题，即“行星轨道的数 目、大小与运动。” 1595年，他终于得到了伟大的发现：“可用地球来量度所有其他轨道。” 他马上着手阐明这一想法，写成了《宇宙的奥秘》初稿。 为了出版这本书，他费尽心机。于是他求救于他的老师。在老师的帮助下， 他这本书终于在1596年面世了，并载入法兰克福书目之中，但署的名却是 “勒普劳斯”。 1598年，由于弗迪南德反对新教教师，开普勒被迫辞去教职。祸不单行， 他的小女儿也不幸夭折。开普勒处于极度的悲愤痛苦之中，于是他只身来到布拉 格。 1600年，开普勒在布拉格结识了天文学家第谷。布拉赫。这是开普勒一 生中最关键的时刻。正是第谷。布拉赫，使开普勒走出逆境，在科学上矗立起一 座丰碑。 由于第谷如此之重要，这里不得不介绍一下第谷的生平。 第谷于1546年生于丹麦斯科纳的一个贵族家庭。13岁时随叔父到哥本 哈根，1562年，他又来到莱比锡。这两个城市的学习为第谷在天文学上的成 就打下了牢固的基础。 第谷被称为是近代天文学的始祖，他的最大贡献是1572年12月11日 发现了仙后星座中的一颗新星，并于1573年发表了题为《新星》的重要科学 论文。 为了完成庞大的天体观测计划，第谷把丹麦国王赠与他的全部补助金，在费 恩岛上建立了有名的福堡天文观象台。 该观象台规模宏大，仪器齐全。这些仪器都是第谷自己设计制造的，有木制 的、铁制的和铜制的。其中最大的是一台精度较高的象限仪，称为第谷象限仪。 由于第谷不断改进仪器的设计和测量的方法，他所进行的大量的天体方位的 测量，其精确度是比较高的，一般能达到半弧分。 第谷在弗恩岛上一直工作了20年之久，除了天体方位的测量外，还发现了 许多新的现象，如黄赤交角的变化、月球的运行的二均差，以及岁差的测定等。 1597年，第谷离开丹麦到汉堡。1599年定居布拉格，并将弗恩岛上 的仪器运到布拉格。1600年，第谷与开普勒会面。从此二人合作开始了新的 工作 10 计划。 开普勒与第谷的会面，乃是欧洲科学史上最重大的事件，这两位个性殊异人 物的相会，标志着近代自然科学两大基础——经验观察和数学理论的有机结合。 也正是这次会合，使开普勒奠定了天体力学的基础和发现行星运动的三大定 律。 1601年，第谷在短期重病后突然离开了人世。第谷临终前对开普勒说： “我一生都在观察星表，我要得到一种准确的星表，我的目标是1000颗星， ……我希望你能把我的工作继续下去。我把我的一切资料全部交给你，愿你把我 观察的结果发表出来，你不会使我失望吧！” 开普勒含泪站在第谷的病床前，沉痛地说：“放心吧，我的老师，我会的！” 开普勒没有使第谷失望，1627年，《鲁道尔夫星行表》便在乌尔姆出版， 第谷的名字永远地载人科学史册。第谷死后，开普勒运用他的大量的观测资料进行细心地研究。当时，不论是 地心说，还是日心说，都认为行星作匀速圆周运动。但开普勒经过深思熟虑，终 于否定了这种长期以来的观点。 他发现火星的轨道是椭圆形的，于是得出开普勒第一定律，即椭圆轨道定律 ：“火星沿椭圆轨道绕太阳运行，而太阳则处于两焦点之一的位置。” 随着火星椭圆形轨道的发现，火星运动的计算也全面展开。开普勒经过计算， 又得出了开普勒第二定律，即相等面积定律：“火星运动的速度是不均匀的，当 它离太阳较近时，运动得较快；反之，则较慢。但从任何一点开始，向经（太阳 中心到行星中心的连线）在相等时间内，所扫过的面积是全部相等的。” 1609年，开普勒的关于火星运动的著作《新天文学》出版。该书还指出 两定律，同样适用于其他行星和月球的运动。这本著作是现代天文学的奠基石。 但开普勒的著作遭到许多人的轻视和误解，开普勒把一切希望都寄托在国外 一个追求真理的人身上，这个人的评价是至关重要的。他就是帕多瓦大学的教授 枷利略。 拉格朗日1736年1月25日生于意大利西北部的都灵。父亲是法国陆军骑兵里的一名军官，后由于经商破产，家道中落。据拉格朗日本人回忆，如果幼年是家境富裕，他也就不会作数学研究了，因为父亲一心想把他培养成为一名律师。拉格朗日个人却对法律毫无兴趣。 到了青年时代，在数学家雷维里的教导下，拉格朗日喜爱上了几何学。17岁时，他读了英国天文学家哈雷的介绍牛顿微积分成就的短文《论分析方法的优点》后，感觉到“分析才是自己最热爱的学科”，从此他迷上了数学分析，开始专攻当时迅速发展的数学分析。 18岁时，拉格朗日用意大利语写了第一篇论文，是用牛顿二项式定理处理两函数乘积的高阶微商，他又将论文用拉丁语写出寄给了当时在柏林科学院任职的数学家欧拉。不久后，他获知这一成果早在半个世纪前就被莱布尼兹取得了。这个并不幸运的开端并未使拉格朗日灰心，相反，更坚定了他投身数学分析领域的信心。 1755年拉格朗日19岁时，在探讨数学难题“等周问题”的过程中，他以欧拉的思路和结果为依据，用纯分析的方法求变分极值。第一篇论文“极大和极小的方法研究”，发展了欧拉所开创的变分法，为变分法奠定了理论基础。变分法的创立，使拉格朗日在都灵声名大震，并使他在19岁时就当上了都灵皇家炮兵学校的教授，成为当时欧洲公认的第一流数学家。1756年，受欧拉的举荐，拉格朗日被任命为普鲁士科学院通讯院士。 1764年，法国科学院悬赏征文，要求用万有引力解释月球天平动问题，他的研究获奖。接着又成功地运用微分方程理论和近似解法研究了科学院提出的一个复杂的六体问题(木星的四个卫星的运动问题)，为此又一次于1766年获奖。 1766年德国的腓特烈大帝向拉格朗日发出邀请时说，在“欧洲最大的王”的宫廷中应有“欧洲最大的数学家”。于是他应邀前往柏林，任普鲁士科学院数学部主任，居住达20年之久，开始了他一生科学研究的鼎盛时期。在此期间，他完成了《分析力学》一书，这是牛顿之后的一部重要的经典力学著作。书中运用变分原理和分析的方法，建立起完整和谐的力学体系，使力学分析化了。他在序言中宣称：力学已经成为分析的一个分支。 1783年，拉格朗日的故乡建立了"都灵科学院"，他被任命为名誉院长。1786年腓特烈大帝去世以后，他接受了法王路易十六的邀请，离开柏林，定居巴黎，直至去世。 这期间他参加了巴黎科学院成立的研究法国度量衡统一问题的委员会，并出任法国米制委员会主任。1799年，法国完成统一度量衡工作，制定了被世界公认的长度、面积、体积、质量的单位，拉格朗日为此做出了巨大的努力。 1791年，拉格朗日被选为英国皇家学会会员，又先后在巴黎高等师范学院和巴黎综合工科学校任数学教授。1795年建立了法国最高学术机构——法兰西研究院后，拉格朗日被选为科学院数理委员会主席。此后，他才重新进行研究工作，编写了一批重要著作：《论任意阶数值方程的解法》、《解析函数论》和《函数计算讲义》，总结了那一时期的特别是他自己的一系列研究工作。 1813年4月3日，拿破仑授予他帝国大十字勋章，但此时的拉格朗日已卧床不起，4月11日早晨，拉格朗日逝世。

以上两个答案有冲突呢，话说詹姆斯。格莱克著有《混沌学》一书，最早是谁引进中国没人清楚，似乎没人对这理论感兴趣，你是从哪里接触的呢？ 就现在来说，似乎只有刘慈欣的《混沌蝴蝶》里提到过