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 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 has a slight change.

If you make the water flow faster, you will get a rhythm of 4 drops, and if the water drops faster, you will get a 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 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 chaos attractor.

Remarks: Actually, what you mentioned is a very famous experiment in chaos theory. 1978, a group of young graduate students from the University of California, Santa Cruz, formed a group to study dynamic systems. When they began to consider the time of the water drop system, they realized that it was not as irregular as it seemed. They use microphones to record the sound of water droplets and analyze the interval sequence between each droplet and the next. They show short-term predictability. If I tell you the falling time of three consecutive drops of water, you can predict the falling time of the next drop of water.

If you are interested in chaos, you can read more relevant books. Chaos is actually a very important theory, which is ubiquitous in our lives.