蝴蝶效应是指在一个动力系统中，初始条件下微小的变化能带动整个系统的长期的巨大的连锁反应。这是一种混沌现象。对于这个效应最常见的阐述是：“一个蝴蝶在巴西轻拍翅膀，可以导致一个月后德克萨斯州的一场龙卷风。” 这句话的来源，是由于这位气象学家制作了一个电脑程序，可以模拟气候的变化，并用图像来表示。最后他发现，图像是混沌的，而且十分像一只蝴蝶张开的双翅，因而他形象的将这一图形以“蝴蝶扇动翅膀”的方式进行阐释，于是便有了上述说法。蝴蝶效应通常用于天气，股票市场等在一定时段难于预测的比较复杂的系统中。此效应说明，事物发展的结果，对初始条件具有极为敏感的依赖性，初始条件的极小偏差，将会引起结果的极大差异。

    The "Butterfly Effect" is the propensity¹ of a system to be sensitive to initial conditions. Such systems over time become unpredictable, this idea gave rise to the notion of a butterfly flapping its wings in one area of the world, causing a tornado² or some such weather event to occur in another remote area of the world.

    Comparing this effect to the domino effect is slightly misleading. There is dependence³ on the initial sensitivity, but whereas a simple linear row of dominoes would cause one event to initiate⁴ another similar one, the butterfly effect amplifies⁵ the condition upon each iteration.

    The butterfly effect has been most commonly associated with the Weather system as this is where the discovery of "non-linear" phenomenon began when Edward Lorenz found anomalies in computer models of the weather. But Henri Poincaré had already made inroads into this area. Mapping the results in "phase space" produced a two-lobe map called the Lorenz Attractor. The word attractor meaning that events tended to be attracted towards the two lobes⁶, and events outside of the lobes are such things like snow in the desert.

    The attractor acts like an egg whisk, teasing apart parameters⁷ that may initially⁸ be close together, this is why the weather is so hard to predict. Super computers run several models of the weather in parallel to discover whether they stay close together or diverge⁹ away from each other. Models that stay similar in nature give an indication that the weather is relatively¹⁰ predictable, and are used to indicate the confidence level that Meteorologists have in a prediction.

    It is not just the weather though that is subject to such phenomena¹¹. Any "Newtonian Classical" system where one system is in competition with another, such as the "Chaotic¹² Pendulum¹³ (混沌钟摆)" which plays magnetism¹⁴ off against gravity will exhibit "sensitivity to initial conditions".

    Animal populations may also be subject to the same phenomena. Work done by Robert May, suggests that predator-prey systems have complex dynamics¹⁵ making them prone¹⁶ to "boom" and "bust”, due to the difference equations that model them. Such a system even with two variables such as Rabbits and Foxes can create a system that is much more complex than would be thought to be the case. Lack of Foxes means that the Rabbit population can increase, but increasing numbers of Rabbits means Foxes have more food and are likely to survive and reproduce, which in turn decreases the number of Rabbits. It is possible for such systems to find a steady state or equilibrium¹⁷, and even though species can become extinct, there is a tendency for populations to be robust¹⁸, but they can vary dramatically under certain circumstances. Real populations of course, have more than two variables making them ever more complex. But as can be seen from the diagram, such systems are not as simple as might be thought.

    The chemical world is also not free from such intrusions of non-linearity. In certain cases chemical feedback produces effects as that in the Belousov-Zhabotinsky reaction (化学混沌反应), creating concentric rings, which are produced by a chemical change, whose decision to change from one state to another cannot be predicted. The B-Z chemical system is currently being trialled as a means to achieve artificially intelligent states in robots.

    Phase space portraits of liquid flow show that they too are subject to the same kind of non-linearity that is inherent in other physical systems. It may be apparent when turning on a tap that sporadic¹⁹ drips become "laminar" as the flow increases. What might not be apparent is the nature of the change from semi-random²⁰ to continuous. It may seem rather at odds²¹ with intuition that such natural systems have inherent behavior that is not random, or indeed that is not capable of being predicted. It may also seem that "not random" means "predictable".

    Natural systems can present a tangled²² mix of determinism and randomness²³, or "order" and "chaos²⁴”. In such cases as water moving from drips to continuous flow, pictures called "Bifurcation diagrams" demonstrate the nature of movement from order into chaos. This bifurcation is based on Robert May's work, but one of the intriguing²⁵ things about bifurcations is that the same pattern occurs no matter what system is iterated. In fact Mitchell Feigenbaum discovered that there was a "constant of doubling" hidden in amongst all these systems.

    Electronic apparatus²⁶ is also not free from such effects, and it is perhaps ironic²⁷, that we think of electronic apparatus as being the epitome²⁸ of predictable determinism and ruthless clockwork efficiency. Indeed the powerful computers used to predict weather, would seem ineffectual if they were not ruthless automatons²⁹. But such effects occur only in certain circumstances where there is "sensitivity to initial conditions”. Amplifiers for instance, produce a howl when feedback occurs as they go into a stable state of oscillation (摆动，震动). Logic³⁰ gates as used in computers have to select a "0" or a "1", and this relies on choosing between two states whose boundary is indeterminate, and it is when a computer confuses a "0" for a "1" or vice³¹ versa that mistakes occur.

    Many of the shapes that describe non-linear systems are fractal, a set of shapes that are self-similar on smaller and smaller scales with no limit to the size of the scale. Fractals were discovered by Benoit Mandelbrot at IBM.

    Fractals have been seen as describing naturally occurring phenomena such as the cragginess of mountains or the shapes of certain plant forms, such as ferns, which can be modeled by affine transformations³².

    Whether in fact Nature is fractal, or whether it just describes it better than the simple geometry of Euclid (欧几里德) depends on the philosophical³³ view taken of mathematics as a whole. Some people think mathematics is just a tool or a creation of man, and therefore Nature is only described or mapped by mathematics.

    Others think that the description is real --- at least in the sense that the similarity is not superficial, that in fact natural objects that look fractal, or which fractals look like, are similar in appearance because at some fundamental level the natural objects are obeying some form of rule system that bears a similarity to the sort of rules which govern fractals.

    Whichever way you look at it, one thing no one can say is that mathematics is irrelevant³⁴ to Nature. From butterflies to plants, from the weather to chemistry, mathematics is modeling or displaying attributes of Nature, and helping³⁵ us to understand what we see.