混沌

Chaos Theory

The Mandelbrot Set

Should a small variation in the force one exerts on the plunger of a pinball machine
be made, then this action can result in a completely different trajectory being
taken by the ball. A butterfly flapping its wings in Beijing could cause heavy
rainfall, instead of sunshine, in New York. Two paper boats placed exactly next
to each other on a river could follow two completely different routes and end
up in two completely different places. These are examples of systems which
display extreme sensitivity in the variation of their initial conditions. Such
dynamical systems are called chaotic, and unpredictability is endemic in them.
However, this is not because these systems are governed by chance. Most of them
can be described by non-linear differential equations, but this non-linear
quality makes predictions and calculations very difficult.

混沌理论

（曼德尔勃特集合）

如果弹球机的活塞中加上某一个微小的力变量，那么这将会导致打出一个完全不同的球的运动轨迹。一个蝴蝶在北京轻轻扇动翅膀，可能在纽约引起一场雨。两个纸船被放置在一条河流的一个地方，哪怕两者靠的非常近，但是两者随水流的路径和最终的位置，都将会相差甚远。这些例子都具有这样的特性：对变量的初始条件非常的敏感。像这样的力学系统就是混沌的，并且不可预知性就是他们的特性之一。但是这并不意味着这些系统仅仅是偶然发生的。绝大多数的这类系统还是可以用非线性的微分方程描述，但是非线性使得预测和计算都变得异常困难。