Mathematical formula for solving chaos

In order to understand how unpredictability is reconciled with determinism, we can look at a system that is far less important than the whole universe-the water drops dripping from the faucet. This is a deterministic system. In principle, the flow of water flowing into the faucet is stable and uniform, and what happens when the water flows out is completely stipulated by the law of fluid movement. But a simple and effective experiment proves that this seemingly deterministic system will produce unpredictable behavior. This gives us some kind of mathematical "lateral thinking", which explains why there are such strange things.

If you turn on the tap carefully and wait for a few seconds, when the flow rate is stable, a series of regular water droplets will usually be produced, which fall at regular rhythm and at the same time interval. It's hard to find anything more predictable. But if you turn on the faucet slowly to increase the water flow and adjust the faucet, a string of water drops will fall in a very irregular way, which seems random. It only takes a few experiments to succeed. Turn the faucet evenly in the experiment. Don't turn it too big, so the water will become an uninterrupted flow. What you need is a medium-speed drip. If you adjust it properly, you can not hear any obvious patterns for many minutes.

From 65438 to 0978, 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. For example, if the last three intervals between water droplets are 0.63 seconds, 1. 17 seconds and 0.44 seconds, it can be determined that the next water droplet will fall after 0.82 seconds (these numbers are for illustration only). In fact, if you know the exact time of the first three drops of water, you can predict the future of the whole system.

So why is Laplace wrong? The problem is that we can never accurately measure the initial state of the system. The most accurate measurement we make in any physical system is correct for about 10 decimal places or 12 decimal places. But Laplace's statement is correct only when we make the measurement reach infinite precision (that is, infinite decimal places, which is of course impossible). In Laplace's time, people have known this measurement error, but it is generally believed that as long as the initial measurement is made, such as 10 decimal, all subsequent predictions will be accurate to 10 decimal. The error neither disappears nor widens. Unfortunately, the error is indeed magnified, which makes it impossible for us to string together a series of short-term forecasts and get a long-term effective forecast. For example, suppose I know the dropping time of the first three drops of water, and the accuracy is 10 after the decimal point, then I can predict the dropping time of the next drop of water, the accuracy is 9 after the decimal point, the accuracy of the next drop of water is 8, and so on. The error is magnified by nearly 10 times every step, and I lose confidence in further decimal places. Therefore, taking a step of 10 into the future, I have no idea about the dropping time of the next drop of water. (The exact number of digits may be different: the precision of 1 decimal place may be lost every 6 drops, but only 60 drops are needed, and the same problem will occur again. )

This error amplification is a logical defect, which shattered Laplace's theory of complete certainty. It is impossible to perfect the whole measurement. If we can measure the dropping time to 100 after the decimal point, our prediction will be invalid when it reaches 100 drops (or 600 drops more optimistically) in the future. This phenomenon is called "sensitivity to initial conditions" or "butterfly effect" (a butterfly flapping its wings in Tokyo may lead to a hurricane in Florida a month later). It is closely related to the highly irregular behavior. By definition, anything that is truly regular is completely predictable. But sensitivity to initial conditions makes behavior unpredictable-and therefore irregular. Therefore, systems sensitive to initial conditions are called chaotic systems. Chaotic behavior satisfies the law of certainty, but it is so irregular that it seems chaotic to the untrained eye. Chaos is not only a complex modeless behavior, but also more subtle. Chaos is a seemingly complex and modeless behavior, which actually has a simple and definite explanation.

The discovery of chaos was made by many people (too many to list here). Its appearance is the confluence of three independent developments. The first is the change of scientific concerns, from simple models (such as repeated cycles) to more complex models. The second is the computer, which enables us to find the approximate solution of the dynamic equation conveniently and quickly. Third, a new mathematical viewpoint about dynamics-geometric viewpoint instead of numerical viewpoint. The first progress provides motivation, the second provides technology, and the third provides understanding.

Geometry of dynamics originated about 100 years ago. Henri Poincare, a French mathematician, was an independent man (if any), but he was so outstanding that many of his ideas became orthodox almost overnight. At that time, he invented the concept of phase space, which is a fictional mathematical space that represents all possible movements of a given dynamic system. For a non-mechanical example, let's consider the group dynamics of a predator-prey ecosystem. In this system, the predator is a pig and the prey is truffles (a strange and spicy fungus). The variables we are concerned about are the size of two populations-the number of pigs and the number of truffles (both relative to a reference value, such as 1 10,000). This choice is actually to make two variables continuous, that is, to take real values with decimal places instead of integer values. For example, if the reference number of pigs is 1 ten thousand, then 17439 pigs is equivalent to the value of 0.0 17439. Now, the natural growth of truffles depends on the number of truffles and the speed at which pigs eat truffles: the growth of pigs depends on the number of pigs and the number of truffles that pigs eat. So the rate of change of each variable depends on these two variables, and we can turn our attention to the differential equation of group dynamics. I won't list equations, because the key here is not equations, but what you do with them.

These equations determine in principle how any initial population value will change with time. For example, if we start with 17439 pigs and 788444 truffles, then you introduce an initial value of 0.0 17439 for the pig variable and 0.788444 for the truffle variable, and the equation will implicitly tell you how these numbers will change. The difficult thing is to make this implication clear: solve the equation. But in what sense do you solve this equation? The natural reaction of classical mathematicians is to find a formula that tells us exactly what the number of pig heads and strains will be at any moment. Unfortunately, such "explicit solutions" are very rare, and it is hardly worth the effort to find them unless the equations have very special and limited forms. Another method is to find approximate solutions on the computer, but that can only tell us what will happen to these specific initial values and what will happen to many different initial values that we want to know most.

Poincare's idea is to draw a picture, drawing all the changes of initial values. The state of the system-the size of two groups at a certain moment-can be expressed as a point on the plane and can be expressed by coordinate method. For example, we can use the abscissa to represent the number of pig heads and the ordinate to represent the number of block strains. The above initial state corresponds to a point with an abscissa of 0.0 17439 and an ordinate of 0.788444. Now let time pass. Coordinates change from one moment to the next according to the rules expressed by differential equations, so the corresponding points will move. Draw a curve according to the moving point; That curve is an intuitive expression of the future state of the whole system. In fact, by observing this curve, you can "see" important dynamic characteristics without knowing the actual value of the coordinates.

For example, if the curve is closed into a loop, the two groups follow a periodic cycle, repeating the same value without interruption, just as cars on the track pass the same audience every lap. If the curve approaches a certain point and stops there, the group will stabilize to a steady state, and nothing will change here-just like a racing car running out of fuel. Due to the coincidence of good luck, cycle and steady state have important ecological significance-in particular, they set upper and lower limits for the size of the population. So these features that are most easily seen by the naked eye are indeed the characteristics of actual things. Moreover, many irrelevant details can be ignored-for example, we can see that there is a closed loop (representing the composite "waveform" of two group periods) without describing its exact shape.

What happens if we try a different pair of initial values? We get the second curve. Each pair of initial values defines a new curve. By drawing a whole family of such curves, we can grasp all possible behaviors of the system at all initial values. This series of curves is similar to the streamline of virtual mathematical fluid hovering around the plane. We call this plane the phase space of the system, and that family of hovering curves is the phase diagram of the system. Instead of the concept of differential equations based on symbols of various initial conditions, we have an intuitive geometric image of points flowing through truffle space. This is different from the ordinary plane only in that many points are potential points rather than real points: their coordinates correspond to the number of pig heads and block strains that may occur under appropriate initial conditions, but they may not appear in certain circumstances. Therefore, in addition to the psychological transfer from symbols to geometry, there is also a transfer from reality to potential philosophy.

For any dynamic system, you can imagine the same type of geometric image. There is a phase space whose coordinates are the values of all variables; Phase diagram, that is, a family of spiral curves representing all possible behaviors from all possible initial conditions, is described by differential equations. This idea is a great progress, because we don't need to care about the exact value of the solution of differential equations, but we can focus on a wide range of phase diagrams, so that people can give full play to their greatest advantage (that is, amazing image processing ability). As a way to weave all potential behaviors (from which nature chooses the behaviors actually observed), phase space diagram has been widely used in science.

Because of Poincare's great innovation, dynamics can be visualized by means of geometric shapes called attractors. If a dynamic system starts from an initial point and observes its long-term operation, it will often be found that it will eventually wander around a certain shape in the phase space. For example, a curve can spiral into a closed loop and then rotate around the loop forever. In addition, different choices of initial conditions will lead to the same terminal shape. If so, this shape is called an attractor. The long-term dynamic characteristics of the system are dominated by its attractor, and the shape of the attractor determines what kind of dynamic characteristics are produced.

For example, a stable system has an attractor, which is a point. Systems that tend to repeat the same behavior periodically have closed-loop attractors. In other words, the closed-loop attractor is equivalent to an oscillator. Please recall the description of vibrating string in Chapter 5: the violin string has undergone a series of movements that finally bring it back to its starting point, and will repeat that series over and over again. I don't mean that the violin string moves in a physical ring, but my description of it is a metaphorical closed loop: the movement travels around in the dynamic terrain of phase space.

Chaos has its peculiar geometric meaning, which is related to the strange fractal shape called strange attractor. The butterfly effect shows that the detailed motion on the strange attractor cannot be determined in advance, but this does not change the fact that it is an attractor. Imagine if you throw an old ball into a rough sea, whether you throw it from the air or let it float from the water, it will move towards the sea. Once it reaches the sea surface, it will go through a very complicated path in the undulating waves, but no matter how complicated the path is, the ball will remain on the sea surface or at least be close to the sea surface. In this picture, the sea surface is the attractor. Therefore, despite chaos, no matter what the starting point may be, the system will eventually be very close to its attractor.

Chaos as a mathematical phenomenon has been fully proved, but how can it be detected in the real world? We have to finish some experiments, but there is a problem. The traditional role of experience in science is to test theoretical predictions, but if the butterfly effect is at work, as it is in any chaotic system, how can we expect to test a prediction? Is chaos essentially untestable and therefore unscientific? The answer is, "no"! Because the word "prophecy" has two meanings. One refers to "predicting the future". When chaos appears, the butterfly effect hinders the prediction of the future. But another meaning is "describe in advance what the experimental results will be." Let's consider the example of tossing a coin 100 times. In order to predict-in the sense of a fortune teller-what will happen, you must list the results of each throw in advance. But you can make scientific predictions, such as "about half of the coins will face up", without having to predict the future in detail-even if you do, the system will still be random. No one will think that statistics is unscientific, because it deals with unpredictable events, so it also treats chaos with the same attitude. You can make all kinds of predictions about chaotic systems. In fact, you can make enough predictions to distinguish deterministic chaos from real randomness. One thing you can often predict is the shape of the attractor, which is not affected by the butterfly effect. The butterfly effect is to make the system follow different trajectories on the same attractor. In a word, the general shape of attractor can often be obtained from experimental observation.

The discovery of mixed tons reveals a basic misunderstanding of the relationship between law and consequential behavior, that is, the relationship between cause and result. We used to think that deterministic causes would produce regular results, but now we know that they can produce extremely irregular results, which is easily misunderstood as randomness. In the past, we thought that a simple reason must produce a simple result (that is, a complicated result must have a complicated reason), but now we know that a simple reason can also produce a complicated result. We realize that knowing these laws does not mean that we can predict future behavior.

How did this disconnect between cause and effect come about? Why does the same law sometimes produce obvious patterns and sometimes mixed oil? The answer can be found in the kitchen of every family, in a mechanical device as simple as an eggbeater. The movements of the two eggbeaters are simple and predictable: each eggbeater rotates smoothly. However, the movement of sugar and protein in the device is much more complicated. Sugar and protein are mixed together under the action of the whisk, which is the function of the whisk, but the two rotating whisks are not twisted together. After that, you don't have to untie the eggbeater arm. Why is the action of stirring egg whites so different from that of the eggbeater arm? Mixing is a dynamic process, which is much more complicated than we thought. Imagine how difficult it is to try to predict where a particular sugar particle will end up! When the mixture passes through a pair of eggbeaters, it is separated left and right. The two sugar grains that were originally together quickly separated and parted ways. This is actually the butterfly effect at work. Small changes in initial conditions have a great influence. Therefore, mixing is a chaotic process.

On the contrary, in Poincare virtual phase space, every chaotic process contains a mathematical mixture. This is why the tides are predictable and the weather is unpredictable. Both of them contain the same type of mathematics, but the dynamics of tides do not mix in phase space, while the dynamics of weather mix in phase space.

Traditionally, science attaches importance to order, but we are beginning to realize that chaos can bring unique benefits to science. Chaos is more likely to respond quickly to external stimuli. Imagine a tennis player waiting to serve. Are they standing still? Do they often move from side to side? Of course not. Their feet are jumping about. Part of the reason is to disturb its opponents; But at the same time, I am ready to respond to any ball that is sent out. In order to move quickly in any particular direction, they move quickly in many different directions. Compared with non-chaotic systems, chaotic systems can easily respond to external events very quickly. This is very important for engineering control problems. For example, we now know that some turbulence is caused by chaos-chaos is the chief culprit in making turbulence chaotic. We may be able to prove that it is possible to establish a control mechanism to respond very quickly to a turbulence that destroys any small area, so that the airflow passing through the surface of the aircraft will not be too fast, thus reducing the motion resistance. In order to respond quickly to the changing environment, organisms must also show chaotic behavior.

This idea has been turned into a very useful practical technology by a group of mathematicians and physicists, including William Tito, Allen garfinkel and Jim York, who call it chaos control. Essentially, the idea is to make the butterfly effect work for you. Small changes in initial conditions lead to great changes in subsequent behaviors, which may be an advantage; One thing you have to do is to make sure that you get the big change you want. Understanding how chaotic dynamics works makes it possible for us to design a control scheme that can fully realize this requirement. This method has achieved some success. One of the earliest achievements of chaos control was that a "dead" satellite changed its orbit and collided with an asteroid, leaving only a very small amount of hydrazine. The National Aeronautics and Space Administration (NASA) maneuvered the satellite to revolve around the moon five times, nudging the satellite with a little hydrazine each time, and finally collided.

This mathematical idea has been used to control the magnetic stripe in turbulence-a prototype used to control the turbulence flowing through submarines or aircraft; Control makes the beating heart return to regular rhythm, which marks the invention of intelligent pacemaker; It is used to establish and prevent the rhythm wave of brain electrical activity, which opens up a new way to prevent epileptic seizures. Chaos has become a rapidly developing industry. Every week, there are new discoveries about the mathematical basis of chaos, new applications of chaos to our understanding of nature, or reports about new technologies of mixing tons, including chaotic dishwashers (an energy-saving machine invented by the Japanese to wash dishes with two chaotic rotating arms) and machines invented by the British to analyze data with chaos theory to improve quality management in mineral water production. However, there is more to be studied. Perhaps the last unsolved problem of chaos is the strange quantum world, and the goddess of luck dominates everything. Radioactive atoms decay with machines, and their only law is statistical law. Although a large number of radioactive atoms have a definite "half-life", when half of them will decay, we can't predict which half will decay soon. Einstein's thesis mentioned above is aimed at this problem. Is there really no difference between decaying radioactive atoms and decaying radioactive atoms? How do atoms know what to do? Is the obvious randomness of quantum mechanics deceptive? Is it really deterministic chaos?

Imagine some kind of vibrating droplet that was originally a cosmic fluid. Radioactive atoms vibrate strongly, and smaller droplets often split and decay. These vibrations are so fast that we can't measure them carefully. We can only measure average quantities (such as energy levels). Now, classical mechanics tells us that a drop of real fluid will vibrate with oil. When it vibrates, its motion is certain, but unpredictable. Many vibrations "randomly" split tiny droplets. The butterfly effect makes it impossible to predict when the droplet will split, but this event has accurate statistical characteristics, including a clear "half-life".

Could the apparent random decay of radioactive atoms be something similar on a microscopic scale? Why on earth is there a statistical method? Statistical law is the expression of inherent certainty. Where will it come from? Unfortunately, no one has let this tempting idea bear fruit-although it is similar in spirit to the popular superstring theory, in which the subgenus particle is a multi-dimensional ring that vibrates artificially. The main similarity here is that both vibrating rings and vibrating droplets introduce new "internal variables" into their physical images, but the significant difference lies in the way they deal with quantum uncertainty. Like traditional quantum mechanics, superstring theory regards this uncertainty as real randomness. However, in a system like a droplet, the apparent uncertainty is actually generated by a deterministic (but chaotic) motive force. The trick-as long as we know how to operate it-may lie in inventing a structure that maintains the successful characteristics of superstring theory and creating several internal variables with chaotic behavior. It may be a touching way to let God's dice determine and Einstein be happy in the spirit of heaven.

What matters is not what you do, but how you do it.

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 some things that we used to think were irregular may actually be the result of simple laws. Natural mixed tons are also bound by law. 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 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 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 number has another symbol: the Greek Umu 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 must turn on the tap by this extra amount and decrease the l/4.669 when each cycle is doubled.

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 opened 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.