米切尔·费根鲍姆Mitchell Feigenbaum

米切尔·费根鲍姆的父亲是马克斯·亚伯拉罕 Joseph Feigenbaum，一位分析化学家，其父母从波兰华沙附近的一个城镇移民到美国。费根鲍姆（或人们所知的Mitch）的母亲是Mildred Sugar，其父母从基辅移民到美国。费根鲍姆是他父母三个孩子中的中间孩子，有一个哥哥Edward和一个妹妹Glenda。

费根鲍姆五岁时进入一所为天才儿童开设的公立学校。与Edward不同，Edward表现出神童的所有特征，从很小就开始阅读，费根鲍姆入学时还不会阅读，他需要母亲的辅导才能达到其他孩子的水平。转到另一所学校后，他变得有些无聊，在其他孩子中没有朋友。事实上，直到上大学之前，费根鲍姆都不喜欢与同学为伴。

费根鲍姆的母亲在他读五年级时教他代数，但阅读仍然是他不太喜欢的事情。也许原因是他尝试阅读Encyclopaedia Britannica中的文章，鉴于他年纪太小，这些文章对他来说太难理解了。十二岁时，他开始在布鲁克林接受高中教育。大约同时，他开始出现某些强迫性倾向，比如过度清洁，这意味着他不停地洗手。这些困难困扰了他好几年，但他在上大学时克服了它们。

学校系统似乎无法为费根鲍姆提供正确的激励，因为他尽管在学业上取得了显著进步，并在覆盖全州的考试中数学和科学获得满分，却尽可能逃避上课。即使他去了布鲁克林享有盛誉的Tilden高中，费根鲍姆也觉得那里的教育并不更令人愉快，尽管他再次在考试中表现出色。

在[1]中，费根鲍姆描述了他对计算的热爱是如何从学校开始的：-

……从初中开始，我决定自己可以计算对数表，后来是三角函数表。我喜欢艾萨克·牛顿解决超越方程的方法，在高中时我就知道起始值可以产生很大差异，并导致非收敛的跳跃，直到手动算术的耐心极限。我父亲在初中时给我看了他漂亮的象牙镶嵌在红木上的计算尺，我很快意识到了它的原理。我被允许使用新的Friden计算器，它在变成遗物之前不久，还可以提取平方根。我喜欢数字，总是作为一种娱乐，更严肃地说，发明新的算法来计算它们。

事实上，在学校期间，费根鲍姆通常通过自学学到的比在正式课程中学到的更多。他大约12岁时就已经自学了弹钢琴，但在高中时他自学了微积分。也是在高中时，他父亲的一个朋友给了他一个带有开关电路的机械装置，可以玩尼姆游戏和其他游戏。这台机器附带了一篇克劳德·香农关于布尔逻辑的论文，这吸引了具有自学态度的费根鲍姆。

1960年2月，十六岁的费根鲍姆进入纽约城市学院。在那里他学习电气工程，但除了电气工程课程外，还参加了所有数学课程和物理课程。他在不到四年内完成了五年课程，于1964年获得学士学位。那年夏天，他开始在麻省理工学院进行研究生学习。他进入麻省理工学院打算研究电气工程以获得博士学位，但仅一个学期后，他转到物理并开始研究广义相对论。

现在，广义相对论又是他自学的一个主题，阅读了列夫·朗道和Evgenii 叶夫根尼·利夫希茨的著作Course of Theoretical Physics。他的正式课程是量子力学、经典力学和复变函数论。正是在MIT期间，费根鲍姆第一次使用了计算机，但不是作为他在那里学习的一部分。当他访问布鲁克林理工学院时，他发现他们有一台可编程数字计算机。他写道[1]：-

这是我使用过的第一台计算机，不到一个小时，我就编程让它用艾萨克·牛顿的方法计算平方根。

费根鲍姆的博士研究由Francis Low指导，他于1970年获得博士学位，学位论文研究的是色散关系。此后，他前往康奈尔大学担任讲师/研究助理，这个职位一半由NSF博士后资助，一半作为教学职位资助。在康奈尔大学的两年里，他教授变分技术和量子力学课程。他在康奈尔使用了一台HP计算机，或许更准确地说是可编程计算器。这台机器只有另一个用户Ken Wilson，因此他能够花时间精通其使用。

在康奈尔大学两年后，费根鲍姆前往弗吉尼亚理工学院担任博士后研究员，同样是一个为期两年的职位。他再次授课，教授斯特凡·巴拿赫空间和$C^{*}$-代数课程。当然，这些短期职位并不理想。正如费根鲍姆所说（见[7]）：-

这些两年期的职位几乎使严肃的工作无法进行。一年后，你就得开始担心下一步能去哪里。

在弗吉尼亚理工学院两年后，费根鲍姆获得了洛斯阿拉莫斯理论部的一个长期职位。他写道[1]：-

当我到达洛斯阿拉莫斯时，理论部主任P Carruthers认为时机合适，而我是合适的人选，来看看Wilson的重整化群思想能否解决这个长达一个半世纪的湍流问题。简而言之，它不能——或者迄今为止还没有——但这引领我走向了奇妙的方向。

费根鲍姆在这里提到的‘奇妙的方向’涉及混沌的研究，他将在那里做出非凡的发现。这一发现之所以成为可能，是因为计算数据可用，而且正如费根鲍姆自己指出的，只是因为他的计算机计算速度太慢，以至于他能看到计算的中间步骤，这才变得显而易见。费根鲍姆与计算机的接触在1974年12月向前迈进，当时他第一次拥有了自己的可编程计算器，HP65。用这台机器[1]：-

我迅速发明了新的常微分方程求解器、最小化程序、插值方法等。对于关心数字的人来说，许多繁琐的工作被消除了。

1976年，时任普林斯顿大学生物学教授的⟦N1⟧爵士指出，逻辑斯蒂映射导致了混沌动力学。逻辑斯蒂映射$g$定义为

$x_{n+1} = g(x_{n}) = \lambda x_{n}(1 - x_{n})$。

它模拟相对种群$x_{n}$，即实际种群与最大种群之比。每次迭代根据旧值给出新的相对种群。参数λ是有效增长率。我们必须有$0 < x_{n} ≤ 1$且0 ≤ λ ≤ 4。

对于 $\lambda < 1, x_{n}$ 趋于 0。对于 $1 ≤ \lambda ≤ 3, x_{n}$ 趋于 $1 - \Large\frac{1}{\lambda}\normalsize$。超过 3 时发生分岔（对应于交替年份的高种群和低种群）。进一步的分岔发生，直到大约 λ = 3.57... 时混沌动力学开始。

1973 年曾有人猜想，逻辑斯蒂方程的行为在定性意义上对于所有具有最大值且在该最大值两侧单调递减的 $g(x)$ 都是相同的。费根鲍姆 获得的非凡结果表明，不仅行为在定性上相似，而且对于所有这样的逻辑斯蒂方程都有一个非常精确的数学结果成立。

费根鲍姆实际上并没有研究May所研究的精确逻辑斯蒂方程，事实上他的工作独立于May的工作。费根鲍姆指出的是，如果我们用上面建立的记号来表述，如果$\lambda_{n}$是发生第$n$次分岔的参数值，那么

$\Large \frac{\lambda_{n} - \lambda_{n-1}}{\lambda_{n+1} - \lambda_{n}}\normalsize \rightarrow 4.669201609102...$当$n \rightarrow \infty$时。

费根鲍姆于1975年8月首次发现4.669，由于他的HP65计算器的精度限制，他只找到三位小数，他花了一些时间试图看看它是否是“众所周知”的数的简单组合。他没有找到任何东西。当然，现在这个数是“众所周知”的，被称为Feigenbaum number。

这本身就令人惊讶，但1975年10月费根鲍姆发现，对于一大类倍周期映射，这个数是相同的。这确实非同寻常，费根鲍姆立刻意识到了它的重要性[1]：-

那天晚上我打电话给父母，告诉他们我发现了一些真正非凡的东西，一旦我理解了它，就会让我成为一个著名的人。

到1976年4月，费根鲍姆完成了关于这个主题的第一篇论文。他把它提交给了一家期刊，但经过六个月的审稿后，他们拒绝了。到1977年，已有超过1000名科学家向他索要副本。他最终在1978年设法将其发表。他的第二篇更技术性的论文于1976年11月完成，遭遇了类似的命运，首次提交时被拒绝。它最终在1979年付印。费根鲍姆在[4]中对非线性动力系统中的倍周期分岔作了初等综述。

费根鲍姆对混沌理论还做出了其他贡献，并且还写了两篇关于地图制作数学的论文。在其中一篇（论文[2]）中，费根鲍姆写道：-

从数字数据库构建地图需要开发许多特殊工具。这些工具包括推广线划和自动放置注记的方法等。此外，鉴于计算机的数值能力及其对绘制线条和圆还是解析上复杂得多的曲线都无所谓，存在一个机会来制作比以前可能的高得多的保真度的投影。因此，人们应该开发工具来利用这种能力并使地图学现代化。……按照档案标准进行的地图学现代化提出了许多问题，非线性系统的思想和方法为这些问题的解决提供了强有力的启示。用这些方法构建的地图都首次出现在恰好一年前出版的⟦E1⟧中。

Hammond Atlas的引言指出[6]：-

数学物理学家费根鲍姆利用分形几何来描述海岸线等自然形态，开发出能够重新配置海岸线、边界和山脉以适应多种地图比例尺和投影的软件。费根鲍姆博士还创建了一个新的计算机化文字放置程序，能在几分钟内放置数千个地图标注，而这项任务以前需要数天的繁琐劳动。

在这一点上，人们或许有理由想知道费根鲍姆认为自己是数学家还是物理学家。他的观点是，物理学与数学之间没有严格的区分。我们同意他的看法，而且在构建这个档案库时，我们确实采取了数学包含理论物理学的观点。

1982年，费根鲍姆在被任命为康奈尔大学教授后离开了洛斯阿拉莫斯。四年后，他成为洛克菲勒大学首位丰田教授。在他被任命到洛克菲勒大学的同一年，他被授予沃尔夫物理学奖。该奖项的授奖词说，此奖授予费根鲍姆：-

……因其开创性的理论研究展示了非线性系统的普遍特征，从而使对混沌的系统研究成为可能。

他获奖时发布的新闻稿很好地总结了他的贡献：-

费根鲍姆的发现产生了非凡的影响。它跨越了理论与实验数学的新领域……很难想到近期理论科学中还有任何其他进展能在如此广泛的领域产生如此广泛的影响，既涵盖非常纯粹的部分，也涵盖非常应用的部分。

在费根鲍姆获得的众多奖项中，我们提及1980年洛斯阿拉莫斯国家实验室的杰出表现奖、1982年美国能源部颁发的欧内斯特·O·劳伦斯奖、1984年的麦克阿瑟基金会奖，以及2005年纽约市市长颁发的科学技术卓越奖：-

……因其在混沌理论方面的开创性研究。

Mitchell Feigenbaum's father is Abraham Joseph Feigenbaum, an analytic chemist whose parents had emigrated from a town near Warsaw in Poland to the United States. Mitchell's (or Mitch's as he is known) mother is Mildred Sugar whose parents emigrated to the United States from Kiev. Mitchell was the middle child of his parents three children, having an older brother Edward and a younger sister Glenda.

Mitchell entered a public school for gifted children when he was five years old. Unlike Edward who displayed all the characteristics of a child prodigy, reading from a very young age, Mitchell could not read when he entered school and he needed tutoring from his mother to bring him up to the level of the other children. Moved to a different school, he became somewhat bored and had no friends among the other children. In fact up until the time he went to university Mitchell would not enjoy the company of his fellow pupils.

Feigenbaum's mother taught him algebra when he was in the fifth form but reading continued to be something that he did not like much. Perhaps the reason was that he tried reading articles in Encyclopaedia Britannica which, given that he was so young, proved too difficult for him to understand. When he was twelve years old he started his high school education in Brooklyn. About the same time he began to develop certain obsessive tendencies such as excessive cleanliness which meant that he was continually washing his hands. He suffered these difficulties for quite a few years but overcame them when a university student.

The school system seemed unable to provide Feigenbaum with the right stimulus for he tried as hard as he could to avoid classes despite making remarkable academic progress and scoring full marks in mathematics and science in the examinations covering the State. Even when he went to Tilden High School in Brooklyn, a school with a fine reputation, Feigenbaum found the education there no more enjoyable, despite once again excelling in examinations.

In [1] Feigenbaum described how his love of calculating started at school:-

... starting in junior high school, I decided that I could calculate the logarithm table myself, and later the trigonometric tables. I loved Newton's method for solving transcendentals, and in high school I already knew that starting values can make a big difference and lead to non-convergent jumps up to the limit of patience of manual arithmetic. My father showed me his beautiful ivory-on-mahogany slide rule in junior high school, and I quickly realised its idea. I was allowed to use the new Friden calculating machine which, shortly before its transformation into a relic, could also extract square roots. I love numbers and always as an amusement, and more seriously than that, invented new algorithms to calculate them.

In fact while at school Feigenbaum had usually learnt more in studying by himself than in the formal lessons. He had already taught himself to play the piano when he was about 12 years old, but at high school he taught himself calculus. Also at high school a friend of his father gave him a mechanical device with switching circuits that could play nim and other games. The machine came with a paper by Shannon on Boolean logic which fascinated Feigenbaum with his self-learning attitude.

In February 1960, at the age of sixteen, Feigenbaum entered the City College of New York. There he studied electrical engineering but attended all the mathematics courses and the physics courses in addition to those in electrical engineering. Completing the five year course in less than four years he graduated with a Bachelor's degree in 1964. In the summer of that year he began his graduate studies at Massachusetts Institute of Technology. He entered MIT with the intention of researching in electrical engineering for his doctorate but after only one term he changed to physics and began to study general relativity.

Now again general relativity was a topic which he studied on his own, reading the book Course of Theoretical Physics by Lev Landau and Evgenii Lifshitz. His official courses were on quantum mechanics, classical mechanics, and complex function theory. It was while he was at MIT that Feigenbaum first used a computer but not as part of his studies there. It was when he was visiting Brooklyn Polytechnic that he found they had a programmable digital computer. He writes [1]:-

This was the first computer I ever used, and within an hour had programmed it to take square roots by Newton's method.

At MIT Feigenbaum's doctoral studies were supervised by Francis Low and he was awarded a doctorate in 1970 for a dissertation on dispersion relations. Following this he went to Cornell as an instructor/research associate, a post which was half funded by an NSF postdoctoral grant, and half funded as a teaching post. During his two years at Cornell he taught courses on variational techniques and on quantum mechanics. He used a HP computer at Cornell which perhaps could be better described as a programmable calculator. The machine had only one other user, Ken Wilson, so he was able to spend time mastering its use.

After the two years at Cornell, Feigenbaum went to Virginia Polytechnic Institute as a postdoctoral worker, again with a two year position. He again taught, giving courses on Banach spaces and $C^{*}$-algebras. Certainly these short term posts were not ideal. As Feigenbaum said (see [7]):-

These two year positions made serious work almost impossible. After one year you had to start worrying about where you could go next.

After the two years at Virginia Polytechnic Institute, Feigenbaum was offered a long term position on the staff of the theory division at Los Alamos. He writes [1]:-

When I arrived at Los Alamos, the theory division head, P Carruthers, felt that the time was right, and I was the appropriate person, to see if Wilson's renormalisation group ideas could solve the century and a half old problem of turbulence. In a nutshell, it couldn't - or so far hasn't - but led me off in wonderful directions.

The 'wonderful directions' that Feigenbaum refers to here involve the study of chaos where he was to make a remarkable discovery. It was made since data was available from computing and, as Feigenbaum himself has noted, only became obvious because the computers he used calculated so slowly that he could see the intermediate steps of the calculation. Feigenbaum's involvement with computers moved forward in December 1974 when he got his own programmable calculator for the first time, the HP65. With this machine [1]:-

In swift order, I invented new ODE solvers, minimisation routines, interpolation methods, etc. For someone who cares for numbers, much of the tedium was eliminated.

In 1976 Sir Robert May, then a professor of biology at Princeton, pointed out that the logistic map led to chaotic dynamics. The logistic mapping $g$ is defined by

$x_{n+1} = g(x_{n}) = \lambda x_{n}(1 - x_{n})$.

It models the relative population $x_{n}$ which is the ratio of the actual population to the maximum population. Each iteration gives the new relative population in terms of the old one. The parameter λ is the effective growth rate. We must have $0 < x_{n} ≤ 1$ and 0 ≤ λ ≤ 4.

For $\lambda < 1, x_{n}$ tends to 0. For $1 ≤ \lambda ≤ 3, x_{n}$ tends to $1 - \Large\frac{1}{\lambda}\normalsize$. Beyond 3 a bifurcation occurs (corresponding to high and low populations in alternate years). Further bifurcations occur until at about λ = 3.57... chaotic dynamics sets in.

In 1973 it had been conjectured that the behaviour of the logistic equation was the same in a qualitative sense for all $g(x)$ which have a maximum value and decrease monotonically on either side of this maximum. The remarkable result obtained by Feigenbaum was to show that not only was the behaviour qualitatively similar but there was a very precise mathematical result which held for all such logistic equations.

Feigenbaum did not actually work with the precise logistic equation which May studied and in fact his work was independent of that by May. What Feigenbaum pointed out, if we state it in terms of the notation set up above, was that if $\lambda_{n}$ is the parameter value at which the $n$th bifurcation occurs then

$\Large \frac{\lambda_{n} - \lambda_{n-1}}{\lambda_{n+1} - \lambda_{n}}\normalsize \rightarrow 4.669201609102...$ as $n \rightarrow \infty$.

When Feigenbaum first found 4.669 in August 1975, which he only found to three places due to the limit of the accuracy of his HP65, he spend some time trying to see if it was a simple combination of 'well-known' numbers. He did not find anything. Of course, now the number is 'well-known' and called the Feigenbaum number.

This in itself was surprising but in October 1975 Feigenbaum found that this number is the same for a large class of period doubling mappings. This was indeed remarkable and Feigenbaum realised the significance of it immediately [1]:-

I called my parents that evening and told them that I had discovered something truly remarkable, that, when I had understood it, would make me a famous man.

By April 1976 Feigenbaum had completed his first paper on the topic. He submitted it to a journal but after taking six months to referee the paper they rejected it. By 1977 he had been asked by over a 1000 scientists for a copy of it. He eventually managed to get it published in 1978. His second, more technical, paper finished in November 1976, suffered a similar fate and was rejected when first submitted. It eventually appeared in print in 1979. Feigenbaum presents an elementary review on period-doubling bifurcations in nonlinear dynamical systems in [4].

Feigenbaum has made other contributions to the theory of chaos and he has also written two papers on the mathematics of making maps. In one of these (the paper [2]) Feigenbaum writes:-

Constructing maps from a digital database requires the development of a number of special tools. These, amongst others, include methods for generalising linework and for the automated placement of type. Additionally, granted the numerical power of a computer with its attendant indifference to whether it plots lines and circles or analytically much more complicated curves, an opportunity exists to craft projections of much higher fidelity than have previously been possible. Thus, one should develop tools to capitalise on this power and modernise cartography. ... The modernisation of cartography done to archival standards poses many problems, the solutions for which are strongly illuminated by the ideas and methods of nonlinear systems. The maps constructed with these methods all appeared for the first time in The Hammond Atlas of the World, published exactly one year ago.

The Introduction to the Hammond Atlas notes [6]:-

Using fractal geometry to describe natural forms such as coastlines, mathematical physicist Mitchell Feigenbaum developed software capable reconfiguring coastlines, borders, and mountain ranges to fit a multitide of map scales and projections. Dr Feigenbaum also created a new computerised type placement program which places thousands of map labels in minutes, a task which previously required days of tedious labour.

It might at this point be reasonable to wonder whether Feigenbaum considers himself a mathematician or a physicist. His view is that there is no hard distinction between physics and mathematics. We agree with him and certainly in constructing this archive we have taken the view that mathematics includes theoretical physics.

In 1982 Feigenbaum left Los Alamos when he was appointed to a professorship at Cornell. Four years later he became the first Toyota professor at Rockefeller University. In the same year that he was appointed to Rockefeller University he was awarded the Wolf Prize in physics. The citation for the prize said that it was awarded to Feigenbaum:-

... for his pioneering theoretical studies demonstrating the universal character of non-linear systems, which has made possible the systematic study of chaos.

The press release made at the time that he was awarded the prize, sums up nicely his contribution:-

The impact of Feigenbaum's discoveries has been phenomenal. It has spanned new fields of theoretical and experimental mathematics ... It is hard to think of any other development in recent theoretical science that has had so broad an impact over so wide a range of fields, spanning both the very pure and the very applied.

Among other awards that Feigenbaum has received we mention the Los Alamos National Laboratory's Distinguished Performance Award in 1980, the Ernest O Lawrence Award by the U.S. Department of Energy in 1982, a MacArthur Foundation award in 1984, and in 2005 the New York City Mayor's Award for Excellence in Science and Technology:-

... for his pioneering studies in chaos theory.