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这几个题目每一个都有很多内容可以挖掘，而且网上的论文多的数不清，建议你到sci上查查，会有很多的，但是pdf格式的，我没法考给你。但是如果你要是为了对付老师混毕业，对不起，没法帮你。
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。
chaos 英音：['kei�0�0s]美音：['keɑs] 混沌 发表在美国物理协会The American Institute of Physics的杂志《混沌》Chaos上的一项 混乱chaos 混乱吵闹可以记做吵死的汉语拼音加记忆 quaint 古怪的怪的汉语拼音guai把q用g 紊乱427 chaos n 混乱紊乱 428 discount n价格折扣 429 display nvt 陈列展览 Chaos 混乱度 Chaos混乱混乱度 Character角色 Character Structures角色结构 Child孩子 浑沌怪 72 Chaos 浑沌怪 73 Dark Awakening 五面怪的诡计 74 Forever Is a Long Time Coming
论文的参考文献的写法，各个学术期刊的要求是不同的，你查阅一下你准备投稿的期刊的参考文献的格式，按照格式的要求写就可以了。如果是初次写的话，请仔细的阅读参考文献的格式，特别注意一些细节的写法，写好了再仔细的核对一下，包括半角、全角符号的使用等。
A连续出版物〔序号〕 主要责任者．文献题名〔J〕．刊名，出版年份，卷号(期号)：起止页码．〔1〕 袁庆龙，候文义．Ni-P合金镀层组织形貌及显微硬度研〔J〕太原理工大学学报，2001，32(1)：51-53B专著〔序号〕 主要责任者．文献题名〔M〕．出版地：出版者，出版年：页码． 〔2〕 刘国钧，郑如斯．中国书的故〔M〕北京：中国青年出版社，1979：115． C会议论文集〔序号〕 析出责任者．析出题名[A]．见(英文用In)：主编．论文集名[C]．(供选择项：会议名，会址，开会年)出版地：出版者，出版年：起止页码．〔3〕孙品一．高校学报编辑工作现代化特征〔A〕．见：中国高等学校自然科学学报研究会．科技编辑学论文集(2)[C]．北京：北京师范大学出版社，1998：10-22．D专著中析出的文献〔序号〕 析出责任者．析出题名[A]．见(英文用In)：专著责任者．书名[M]．出版地：出版者，出版年：起止页码．〔4〕罗云．安全科学理论体系的发展及趋势探讨[A]．见：白春华，何学秋，吴宗之．21世纪安全科学与技术的发展趋势[M]．北京：科学出版社，2000：1-5．E学位论文[序号] 主要责任者．文献题名[D]．保存地：保存单位，年份：[5〕张和生．地质力学系统理论[D]．太原：太原理工大学，1998：F报告[序号] 主要责任者．文献题名[R]．报告地：报告会主办单位，年份： [6]冯西桥．核反应堆压力容器的LBB分析[R]．北京：清华大学核能技术设计研究院，1997：G专利文献〔序号〕 专利所有者．专利题名〔P〕．专利国别：专利号，发布日期：〔7〕姜锡洲．一种温热外敷药制备方案〔P〕．中国专利：881056078，1983-08-12：H国际、国家标准〔序号〕 标准代号．标准名称〔S〕．出版地：出版者，出版年：〔8〕GB/T 16159—1996．汉语拼音正词法基本规则〔S〕．北京：中国标准出版社，1996：I报纸文章〔序号〕 主要责任者．文献题名〔N〕．报纸名，出版年，月(日)：版次〔9〕谢希德．创造学习的思路[N]．人民日报，1998，12(25)：10J电子文献〔序号〕 主要责任者．电子文献题名〔文献类型/载体类型〕．：电子文献的出版或可获得地址(电子文献地址用文字表述)，发表或更新日期/引用日期(任选) ：〔10〕姚伯元．毕业设计(论文)规范化管理与培养学生综合素质〔EB/OL〕．：中国高等教育网教学研究，2005-2-2：
根据学术论文的要求，投稿需要具有新的科学研究成果或创新见解和知识的科学记录；或是某种已知原理应用于实际并取得新进展的科学总结。应提供新的科技信息，内容应有所发现、创造和前进，而不是重复、模仿和抄袭。《学刊》投稿范围包括：城市规划理论和概念、规划政策和实践、城市规划技术与方法、城市空间发展问题与规划、规划师的社会作用，以及相关学科的交叉研究等。来稿务求原创性，稿件未投它刊，未在其它任何期刊(包括内部出版物)刊登或研讨会上发表过(特约除外)。本刊接受doc文件格式和可编写的pdf文件格式文稿，投稿时请登陆本刊网站上传稿件，或将稿件和电子文件一同寄（送）到本刊编辑部，也可以用电子邮件把文稿以附件形式发送至本刊的电子邮箱。文中插图另附if格式的图形文件，分辨率在300dpi以上。本刊不收版面费，文稿刊出后，编辑部向作者赠送2本当期期刊，并酌付稿酬。 投稿的编写格式一般包括前置部分和主体部分，必要时还有附录部分。前置部分包括题名、作者、摘要和关键词。主体部分包括引言、正文、结论、致谢和注释、参考文献。1 题名、作者题名是以最恰当、最简洁的词语反映论文中最重要的特定内容的逻辑组合。一般不超过20字。作者附简介信息，包括工作单位、职务、职称，电子邮件地址等。如系基金资助项目、重大科研项目的论文，请在稿件首页页脚位置注明基金名称、项目全称和编号。2 提要与关键词提要是论文内容不加注释和评论的简短陈述。具有独立性和自含性，有数据有结论。一般说明研究目的、方法、结果和最终结论等，重点是结果和结论。中文提要一般300字左右。关键词是从论文中选取出来的用以表示全文主题内容款目的单词或术语，一般3～8个。投稿同时需要提供英文题名、英文提要和关键词。3 标题、图、表的序号文章序号以阿拉伯数字分级编号，如：1，1，1，一般为三级编号，不超过五级。图、表分别以阿拉伯数字顺序编号，如：表1，表2；图1，图2。图和表都应有确切简短的题名，图名连同图号置于图下；表名连同表号置于表上。在正文中表，图及其编号要提及并对应。4 单位采用国际单位制和可与国际单位制单位并用的中国法定计量单位。5 参考文献表参照GB/T 7714-2005，《城市规划学刊》常用参考文献类型的格式依次示例如下：1连续出版物[1] 刘武，郑良, 姜础 元谋古猿牙齿测量数据的统计分析及其在分类研究上的意义[J]科学通报, 1999,44[23]: 2481～[2] 沈福煦 中国古代文化的建筑表述[J] 同济大学学报:人文社会科学版，1997 [2]：1~[3]Hewitt JA Technical Services in 1983[J] Library Resource Services, 1984,28(3):205～2专著[1]余敏 出版集团研究[M] 北京：中国书籍出版社，2001：179～[2] Marcus, CC & Sarkissian, W, Housing as if People Mattered[M] University of California Press, Berkeley, [3]Mumford, L, 城市发展史—起源、演变和前景[M] 倪文彦，宋峻岭，译 中国建筑工业出版社，Crawfprd W, Gorman M Future Libraries: Dreams, madness, & reality [M] Chiacago:American Library Association,3 专著中析出文献[1]腾序汉唐西域古城研究[M]//董鉴泓主编 城市规划历史与理论研究 同济大学出版社，1999：96~[2]马克思 关于《工资、价格和利润》的报告札记[M]//马克思, 恩格斯 马克思恩格斯全集: 第44卷 北京: 人民出版社，1982: [3]Buseck P R, Veblen DR, Subsolidus Phenomena in Pyroxenes[M]//Prewitt C TP Washington, DC : Mineralogical Society of Aerica,1980: 117～4 学位论文与会议论文[1]孙玉文 汉语变调构词研究[D] 北京: 北京大学, [2]辛希孟 信息技术与信息服务国际研讨会论文集: A集[C]北京: 中国社会科学出版社, 5 报纸[1]胡必亮 小城镇大战略[N] 光明日报，1999-04-23(6)6 电子文献[1]傅刚, 赵承, 李佳路 大风沙过后的思考[N/OL] 北京青年报,2000-04-12(14) [2005-07-12][2] Turcotte D L, Fractals and Chaos in Geology and Geophysics [M/OL] New York: Cambridge University Press, 1992[1998-09-23]附：文献类型和标志代码：普通图书(M)、会议(G)、报纸(N)、期刊(J)、学位论文(D)、报告(R)、标准(S)、专利(P)、数据库(DB)、计算机程序(CP)、电子公告(EB)、联机网络(OL)。 作者文责自负，并不代表本刊的观点和立场。本刊有权在不违反作者基本观点的基础上对稿件进行编辑、删改，必要时需请作者修改，不同意删改者请在来稿时注明。请勿一稿多投，收到本刊电子回复3个月后未接到录用通知，可自行处理。限于人力和财力，来稿一律不退，请自留底稿。
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。
什么是混沌[转贴]techana财经论坛 (2001-12-23 21:20:26) --------------------------------------------------------------------------------什么是混沌？[Techana注：一篇介绍混沌的文章，很少见。推荐给大家，尽管我不同意其中的部分观点]混沌一词，来源于英文的chaos[Techana注：KAO，就是中国的“道”，这些翻译猪，连这个都不知]，近些年来除了受到数学、物理学等学术研究领域的关注外，在音乐、艺术、美工设计等方面的应用更加普遍。采用计算机作图技术，根据混沌等式可以画出奇妙无比的图形。例如根据 Z5作出的图形看起来就像蚂蚁，这里Z=5+2SQRT(-1)。 [Techana注：不懂！]20世纪初期法国人路易斯对股票价格这种特殊的运动非常感兴趣，那时他甚至就提出了T5法则，说明股价运动也是一种混沌现象。那么到底什么是混沌呢？[Techana注：先研究“饺子”。：）] 最近见到一本《混沌操作法》[Techana注：一定要读]的书。一些同好的读者认为这是一本市场人士不可不读的书，书中提出了许多崭新的观点。而另外一些读者朋友则认为，它不过是在一个新的名词“混沌”之下重新阐述了波浪原理而已[Techana注：非也非也,绝非如此！]。读书的心得，当然是仁者见仁，智者见智，不必追究。但是由此再次激起了笔者的兴趣：到底什么是混沌现象？所谓的市场混沌操作法究竟是怎样操作的？ 一、拉普拉斯宇宙论 在19世纪，法国的天文学家和数学家拉普拉斯提出：如果知道某种事物的最初状态，那么就可以事先确定它久远的未来状况。[Techana注：Yehhhhhhhhhhh!]他认为，如果人们有足够的智慧把握宇宙万物在某个时间的状况，那么就可以把握它的过去和将来。这就是著名的拉普拉斯宇宙论之基础。[Techana注：我们的智慧？足够吗？我们只要知道有关股票价格的将来就可以]由此我们很容易联想起《旧约圣经·传道书》中著名的一段话：“一代过去，一代又来，地却永远长存。日头出来，日头落下，急归所出之地。风往南刮，又向北转，不住地旋转，而且返回转行原道。江河都往海里流，海却不满；江河从何处流，仍归还何处。已有的事，后必再有；已行的事，后必再行。日光之下，并无新事。”[Techana注：好书哦，不比《道德经》差] 后来对天体运行的观察和研究表明，情况好像不完全是这样。观察的最初条件发生微小的变化都会导致最终结果的巨大差异。因此，预测，尤其是长期预测变成了不太可能的事情。对于具有不确定性的系统或者是对于混沌系统而言，情况更是如此。[Techana注：是的，如果不是当初的一次偶遇，就不会有现在这个小T；相信小T一定有，可是此小T非彼小T也] 二、力学系统的线性特性 古典的力学系统具有线性特性，变量之间存在一定的比例关系。例如，小贝贝的身高每年长高6厘米，可以表述为： x(n+1)=x(n)+6 如果小贝贝今年是80厘米高，即x(n)=80，那么明年就是x(n+1)=80+6=86，即86厘米高。这就是一个典型的具有确定性的力学系统，变量是一次方，因此是线性的。 再例如现代证券投资理论中著名的资本资产定价模型(CAPM)： E( R )=α+β(Rm) 表明市场中存在风险-回报交易，风险是由贝塔值定义的，回报是与风险成正比例关系。 三、混沌系统的特性 首先，混沌系统与古典的力学系统不同，它具有非线性特性。此外通过下例可以看到，混沌系统对于初始条件非常敏感。例如： x(n+1)=4x(n)[1-x(n)] x(n)可以看成是系统输入，x(n+1)可以看成是系统输出，因为等式右边出现了输入变量的平方，因此该等式是非线性的。正是由于等式的这种非线性特性，使得它对于初始条件非常敏感。 假设x(n)=75，则x(n+1)=4(75)[1-75]=75，即x(n+1)=x(n)。 如果这是一个描述市场价格变化的等式，那么市场就会处于平衡。今天的价格是75，产生的明天的价格仍然是75。75这个数值就称之为这个等式的不动点。75是一个不动点，这个等式还有其它不动点吗？所有不动点的集合能够确定吗？经常答案是无法确定的。 假设市场价格以7499开始，即x(0)=7499，则随后的第一个和第二个交易日的价格为： x(1)=4(7499)[1-7499]=7502 x(2)=4(7502)[1-7502]=7496 表1列出了分别以x(0)=75、x(0)=7499和x(0)=74999为初始条件，前20次计算的结果。以第20次的计算结果为例，如果x(0)=75，那么x(20)=75。如果x(0)=7499，那么x(20)=359844。如果x(0)=74999，那么x(20)=995773。很明显，初始值的微小差别在经过几次计算之后就会产生有较大差别的结果。因此，这个等式对于初始条件非常敏感。 表1 不同初始值的前20次计算结果 四、混沌系统说明了什么？ 混沌系统说明简单的确定性系统可以产生看起来是随机的过程。可以从两个方面理解。从便利的一方面来讲，如果我们观察到的是很复杂的现象，也许产生它的却是一些具有确定性的规则。这样，也许我们能够发现它究竟是什么，也许生活根本就不是那么复杂！从不利的一方面来讲，假设我们有一个非常简单的系统，也许我们认为自己已经理解它了——它看起来是那么简单！但是它也许会产生非常复杂的现象。在两种情况下，混沌特性都告诉我们，究竟一个看起来是随机的过程是真正随机的？或确定的？是无法确定的。那么对于股票、期货、利率这样的一些变量来说，究竟是真正的随机变量还是可确定的？这一问题的答案本身就无法确定。[Techana注：本文作者对“混沌”的理解不深刻。“混沌”和“道”有相似之处，都有“混乱中的秩序”的含义，老子也是因为如此，才有“道（混沌）可道（可以说出来的），非常（就不是）道（混沌的本意）”的精论。“混沌”也好，“道”也好，决没有把你搞晕的意思] 我们知道，在过去几十年中，证券投资理论方面明显地分为两大流派，即随机漫步的学院派和市场（技术）分析的市场派，前者认为市场价格是随机的，无法预测的，而后者认为价格是有重复再现规律的，不是随机的。有兴趣者不妨参考《漫游华尔街》。如果认为市场是一个混沌系统，那么我们只好说，价格是否是随机的，这个问题同样是不确定的。[Techana注：自古以来，就是这“派”那“派”的害人啊！追求真理，殊途同归，何来“此派”“彼派”？教唆争斗，是要被判刑的哦。] 看似复杂的问题不一定真正复杂，看似简单的问题未必真正简单。就连这个问题是复杂还是简单本身都无法确定，更何况问题的答案！但是混沌系统带来的也并不完全是悲观。 [Techana注：混沌应该属于哲学的范畴，绝非一个简单的答案]五、混沌特性的作用 历史上，士兵们过桥时整齐的步伐曾经带来桥梁共振，使桥梁倒塌。相反，混沌特性可以使桥梁各个部分的作用相互独立，避免这种现象的发生。[Tehcana注：胡说！经典物理有经典物理的适用范围，混沌有混沌的适用范围。不能因为想引人注目，就标新立异，胡言乱语。你让部队齐步过桥，看看结果] 经济体系中的混沌特性本身也是很有益的，在国际商业循环中可以防止许多国家的经济同时下跌。否则，各国的商业循环也许就会变得比较和谐，这并不一定是件好事。它意味着许多经济实体可能会同时走入低谷。因此国际上过于紧密的经济联合体的出现也许最终会削弱世界经济抗冲击能力。为了生存，自然界需要各种各样的动植物共存，共同维持生态平衡。为了世界的和平，需要各种国际势力的存在，才能够互相制约。同样，只有“混沌”的证券市场才有存在和发展的空间。和谐可以产生美，然而混沌才是和谐赖以开花结果的沃土。[Tehcana注：hehe，和谐和残缺都是美] 混沌系统由于对初始条件极为敏感，看起来根本不可能消除干扰。但是事实上能够非常快地消除干扰。换句话说，正是因为混沌系统本身对于初始条件极为敏感，初始条件本身很快就变得不那么重要了。难怪人们要赞美证券市场这个平等的竞争场所，在那里你还能够说世袭的财富和权势有多少持久的效力吗？[Tehcnana注：小心了，你的喷嚏，可能引起巴西的风暴] 六、预测失效的速度 初始条件的微小差别使得经过几次计算之后结果大幅度发散，那么到底这种发散速度有多快？这是对我们预测能力的衡量。Lyapumov指数λ是衡量计算结果发散的一种方法，它表明预测按照指数速度失效。[Techana注：不懂那个“入”是什么东东] --------------------------------------------------------------------------------i社区原文：什么是混沌[转贴]
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
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。
首字母不同,这你都看不出来，其实没什么
1972年12月29日，美国麻省理工学院教授、混沌学开创人之一EN洛伦兹在美国科学发展学会第139次会议上发表了题为《蝴蝶效应》的论文~从而提出了混沌学理论~ 有关引进中国的问题~ 《礼记·经解》：“《易》曰：‘君子慎始，差若毫厘，缪以千里。’”《魏书·乐志》：“但气有盈虚，黍有巨细，差之毫厘，失之千里。”可见中国在1300年前就已经对于混沌学有相应记载了~所以这个引进人物~不太好说~