Chaos is subverting our comfortable assumptions about the way the world works. On the one hand, chaos tells us that the universe is far weirder than we thought. Chaos makes many traditional scientific methods suspect, and it is not enough to know only the laws of nature. On the other hand, chaos also tells us that something we once thought was irregular may actually be the result of simple laws. Chaos in nature is also restricted by laws. In the past, science often ignored seemingly irregular events or phenomena because they had no obvious pattern at all and were not dominated by simple laws. That was not the case. There are simple laws right under our noses-laws to control epidemics, heart diseases or locust plagues. If we know these laws, we may be able to prevent the ensuing disaster. Chaos shows us new laws, even new laws. Chaos has a new general model. One of the earliest patterns found exists in the drip tap. Maybe we remember that the faucet can drip rhythmically or randomly, depending on the speed of the water flow. In fact, the faucet with regular dripping and the faucet with irregular dripping are slightly different variants of the same mathematical prescription. However, as the speed of water flowing through the faucet increases, the types of dynamic characteristics change. The attractor in the phase space that represents the dynamic characteristics is constantly changing-it changes in a predictable but extremely complicated way.

The faucet that drips regularly has the rhythm of repeated dripping, and each drop is the same as the previous one. Then gently unscrew the faucet, and the water drops a little faster. Now the rhythm becomes drop by drop, repeating every 2 drops. Not only is the size of the water drop (which determines the sound of the water drop), but also the dripping time from one drop to the next is slightly different.

If you make the water flow faster, you get the rhythm of 4 drops, and the water drops faster, and the result is the rhythm of 8 drops. The length of the repeated sequence of water droplets is constantly doubling. In the mathematical model, this process continues indefinitely, and the rhythm groups of water droplets such as 16, 32, 64, etc. However, it is becoming more and more subtle to produce a flow rate that doubles every continuous period; And with a flow rate, the size of the rhythm group will double infinitely frequently. At this moment, no water droplet sequence completely repeats the same pattern. This is chaos.

We can use Poincare's geometric language to express what happened. For the faucet, the attractor is a closed loop at first, indicating a periodic cycle. Imagine that this ring is a rubber band on your finger. When the flow rate increases, the ring splits into two adjacent rings, just like a rubber band winding twice on a finger. So the rubber band is twice the original length, so the cycle is twice as long. Then this doubled ring doubles in exactly the same way along its length, resulting in a cycle with a period of 4, and so on. After turning it over an infinite number of times, your finger is wrapped with a rubber band like spaghetti, which is the chaotic attractor.

This chaotic creation scheme is called period doubling cascade. 1975, physicist Mitchell Feigenbaum discovered that a special number that can be measured through experiments is associated with the multiplication cascade of each period. This number is about 4.669, which, together with π, ranks among the strange numbers that seem to have unusual significance in mathematics and its relationship with nature. Feigenbaum numbers also have a symbol: the Greek letter δ. The number π tells us the relationship between circumference and diameter. Similarly, feigenbaum number δ tells us the relationship between water droplet circulation and water velocity. To be precise, you have to turn on the tap, and this extra amount will be reduced by 1/4.669 in each cycle of doubling.

π is the quantitative characteristic of anything related to a circle. Similarly, feigenbaum number δ is a quantitative feature of any periodic multiplication cascade, no matter how the cascade is generated or how it is obtained through experiments. This number will also appear in experiments on liquid ammonia, water, circuits, pendulums, magnets and vibrating wheels. It is a new universal model in nature, a model that we can only see through the eyes of chaos, a quantitative model generated from qualitative phenomena, and a number. This number is indeed one of the natural numbers. Feigenbaum number opens the door to the new world of mathematics that we have just begun to explore? This precise pattern (harmony and other patterns) discovered by feigenbaum is a masterpiece. The most fundamental point is that even if the results of natural laws seem to have no patterns, laws still exist, and so do patterns. Chaos is not random, it is a seemingly random behavior produced by precise laws. Chaos is the secret form of order.