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In the 1970s, there were two topics of widespread concern in the field of mathematics, one is chaos theory, that is, the so-called nonlinear dynamics. This topic developed from calculus. The other is a complex system, which has a less orthodox way of thinking and stimulates new mathematics and new science.

Chaotic Newton's philosophy of natural mathematical principles simplifies the system of the world to differential equations, which are "deterministic". In other words, once the initial state of the system is known, its future is always the only certainty. This is a theory that leaves no leeway for free will. But at the same time, it also brings us radios, televisions, radars, mobile phones, commercial aircraft, communications satellites and computers.

The development of scientific determinism is accompanied by a vague but deep-rooted concept of conservation of complexity. This is the hypothesis that "simple reasons must produce simple results", and it also means that complex results must have complex reasons. This concept makes us wonder where the complexity of complex systems comes from. For example, since life must have originated from an inanimate planet, where does its complexity come from? We rarely think that complexity may occur automatically, but that's what mathematics shows.

The certainty of the laws of physics from Laplace to Poincare comes from a simple mathematical fact: under a given initial condition, the differential equation has at most one solution. Laplace summed up the mathematical view of determinism:

If this intelligence is strong enough, it can condense the motion of the largest object in the universe and the motion of the lightest atom into a formula. For such an intelligence, nothing is uncertain, and the future appears in front of it like the past.

Ironically, it is in astromechanics (the most obvious determinism part of physics) that Laplace's determinism is challenged. In 1886, King Oscar II of Sweden established an award for solving the problem of the stability of the solar system. Will the solar system operate steadily all the time? Or will a planet crash into the sun? The physical laws of conservation of energy and momentum do not prevent these two possibilities. Can the detailed dynamics of the solar system provide more clues?

Determined to challenge this problem, Poincare carefully studied a simpler problem, that is, the system of three celestial bodies. The equation of three objects is not much worse than that of two objects, and the general form is roughly the same. However, Poincare's three-body problem was unexpectedly difficult, and he found something disturbing. The solutions of these equations are completely different from those of the two-body equation. In fact, these solutions are so complex that they cannot be written down in mathematical formulas. To make matters worse, he knows enough about the geometry (or, more accurately, the topology) of the solutions to prove beyond doubt that the motion represented by these solutions can sometimes be highly disordered and irregular. This complexity is now seen as a classic example of chaos.

About 60 years later, the study of the three-body problem triggered a revolution that changed the way we look at the relationship between the universe and mathematics.

In 1926, Balthazar van der Bohr, a Dutch engineer, built a mathematical model to simulate the heart and found that under some conditions, the oscillations produced were not as periodic as the normal heartbeat, but irregular. Later, mathematicians provided a solid mathematical foundation for van der Bohr's research in a study of radar electronics.

In the early 1960s, the American mathematician Stephen Smeyer demanded a complete classification of the typical behavior types of electronic circuits, thus creating a modern era of dynamic system theory. At first, he thought the answer was a combination of periodic movements, but soon realized that more complex behaviors were possible. In particular, he developed the discovery of Poincare's complex motion in limiting the three-body problem, simplified the geometric structure and produced a system called Smeer's Horseshoe. He proved that although the horseshoe system is definite, it also has some random characteristics.

A universal theory (chaos theory) began to emerge. At the same time, chaotic systems appear sporadically in the applied literature. One of the most famous chaotic systems was proposed by meteorologist Edward Lorentz in 1963. Lorentz is going to establish an atmospheric convection model, using a simpler equation with three variables to approximate the very complex equation of this phenomenon. When using numerical methods to solve these problems on the computer, he found that the solutions oscillated in an irregular and almost random way. He also found that if there was a slight disturbance in the initial value of the variable, the result would be very different. His description of this phenomenon in a subsequent lecture led to the famous "butterfly effect".

This effect brings great problems to the weather forecast. But it would be wrong to conclude that butterflies caused the hurricane. In the real world, it is not a single butterfly that affects the weather, but the statistical characteristics of trillions of butterflies and other small disturbances. In general, these factors have a clear impact on the location, time and subsequent direction of hurricanes.

Using the topological method, the researchers proved that the singular solution observed by Poincare is the inevitable result of the singular attractor in the equation. A strange attractor is a complex motion. It can be visualized as a shape in the state space formed by variables that describe the system.

The structure of attractors explains a peculiar feature of chaotic systems: they can be predicted in the short term, but not in the long term. Why can't several short-term forecasts be strung together to form a long-term forecast? Because the accuracy of our description of chaotic systems decreases over time, and the rate of decline continues to increase, there is a prediction range that we cannot go beyond. In spite of this, the system is still on the same strange attractor, but its path through the attractor has changed significantly.

This has changed our view of the butterfly effect. All butterflies can do is drive weather changes on the same strange attractor.

David Ruel and Flores Tarkens soon discovered the application of strange attractors in physics, the puzzling problem of fluid turbulence. The standard equation of fluid flow, called Navier-Stokes equation, is a partial differential equation. A common fluid flow, laminar flow, is smooth and regular, which is exactly what you expect from deterministic theory. But another type, turbulence, is irregular, almost random. Previous theories either claim that turbulence is a combination of extremely complex modes, each of which is very simple, or that the Navier-Stokes equation fails in the turbulent state. But Ruel and Tarkens have a third theory. They believe that turbulence is a physical example of a strange attractor.

At first, people were skeptical of this theory, but we now know that it is correct. Other successful applications followed, and the word "chaos" was used as the name for all such behaviors.

Theoretical monsters between 1870 and 1930, a group of maverick mathematicians invented a series of strange shapes with the sole purpose of proving the limitations of classical analysis. In the early development of calculus, mathematicians assumed that the amount of any continuous change had a clear rate of change almost everywhere. For example, an object that moves continuously in space has a definite speed. In 1872, however, Weierstras proved this long-standing assumption wrong. An object can move in a continuous way, but its way of movement is very irregular, and its speed is suddenly changing all the time. This means that it doesn't actually have a reasonable speed at all.

The strange shapes found by mathematicians include a curve that fills the entire area of space (one by Piano in 1890 and another by Hilbert in 1891), a curve that crosses at every point, and an infinite curve that surrounds a finite area. The last weird geometric figure, invented by von Koch in 1906, is the snowflake curve, and its structure is like this.

Because of its six-fold symmetry, the final result looks like a complex snowflake.

The mainstream of mathematics began to denounce these strange things as "morbid" and "monster galleries", but over time, the Trinity view was supported. By the turn of the century, mathematicians had begun to accept the "monsters" in the Monster Gallery. By 1900, Hilbert even called the field paradise.

In the 1960s, to everyone's surprise, the gallery of theoretical monsters got an unexpected boost in the direction of applied science. Mandelbrot realized that these strange curves were clues to the theory of irregularities in nature. He renamed them fractals. Nature is full of complex and irregular structures, such as coastlines, mountains, clouds, trees, glaciers, river systems, waves, craters and cauliflower. We need a new natural geometry.

Today, scientists have incorporated fractals into their normal way of thinking. Atmospheric flow is turbulence, turbulence is fractal, and fractal can move continuously like Weirstras's abnormal function, but there is no definite velocity. Mandelbrot has found examples of fractals in many fields both inside and outside science-the shape of trees, the branching patterns of rivers, the movements of the stock market.

Chaos is everywhere in strange attractors. Geometrically, it turns out to be fractal. These two ideological lines are intertwined to form the well-known chaos theory.

Chaos exists in almost every field of science. Mathematicians have found that the dynamics of the solar system is chaotic. Due to the existence of dynamic chaos, the prediction range of the solar system is about 1000 million years. So, if you want to know which side of the sun the earth is on in 1000 AD, it is impossible. These astronomers also show that the tides of the moon stabilize the earth, making the climate livable; so chaos theory shows that without the moon, the earth would be a very uninhabitable place.

Chaos occurs in the mathematical models of almost all biological populations. Ecosystems do not usually reach some kind of static equilibrium of nature. on the contrary, they hover on strange attractors and usually look quite similar, but they are always changing.

Complexity many of the problems facing science today are extremely complex. To manage coral reefs, forests or fishing grounds, it is necessary to understand a highly complex ecosystem in which seemingly harmless changes can cause unexpected problems. The real world is so complex and difficult to measure that traditional modeling methods are difficult to establish and even more difficult to verify. To meet these challenges, more and more scientists are beginning to believe that there is a need for fundamental changes in the way we simulate the world.

In the early 1980s, George Cowen realized that one way forward lies in the newly developed theory of nonlinear dynamics. Before nonlinear dynamics became the main method of scientific modeling, its function was mainly theoretical. The most profound work is Poincare's study of the three-body problem of celestial mechanics. This predicts the high complexity of celestial orbits, but little is known about them. This proves that a simple equation may not have a simple solution, and the complexity is not conserved, but it can have a simpler origin.

Cellular automata in a mathematical model called cellular automata, biology is simplified to colored squares. John von Neumann was trying to understand life's ability to replicate itself. Imagine a universe made up of huge checkered grids called cells, like a giant chessboard. At any time, a given square can exist in a certain state. The chessboard universe has its own natural laws that describe how the states of each cell change over time and use colors to represent those states. The natural law of the chessboard universe is simple: if a cell is red with two blue cells next to it, it must turn yellow. Any such system is called cellular automata, the cell is because of the grid, the automaton is because it blindly obeys any rules listed.

In order to simulate the most basic characteristics of biology, von Neumann created a cellular structure that can replicate itself. It has 200000 cells and carries its own coded description in 29 different colors. This description can be copied blindly. Von Neumann did not publish his research until 1966, when Crick and Watson had discovered the structure of DNA and it became clear how life was actually replicated. Cellular automata have been ignored for another 30 years.

However, in the 1980s, people became more and more interested in systems made up of a large number of simple components, which interact to produce a complex whole. Traditionally, the best way to model a system mathematically is to include as many details as possible. But this high-detail approach is a failure for very complex systems. For example, suppose you want to know about the growth of the rabbit population. You don't need to simulate the length of rabbit fur, how long their ears are, or how their immune system works. You only need to know some basic information about each rabbit, its age, sex, whether it is pregnant or not. Then you can focus your computer resources on what really matters.

Complex systems support the view that life is possible as long as chemistry is complex enough.

Cellular automata are very effective for this kind of systems. They can ignore unnecessary details about individual components and focus on how these components relate to each other.

Geology and biology-the formation of complex systems such as river basins and deltas that cannot be analyzed by traditional modeling techniques. Peter Burrough uses cellular automata to explain why these natural features form the shapes they have. Automata simulate the interaction between water, land and sediment. The results explain how different soil erosion rates affect the shape of rivers and how rivers take away soil, which are important issues in river engineering.

Another important application of cellular automata occurs in biology. Stuart Kaufman applied a variety of complex theoretical techniques to delve into another major problem in biology: the development of organic forms. The growth and development of an organism must involve a large number of dynamic processes, and it is not just a matter of transforming the information stored in DNA into organic form. An effective method is to express the development as the dynamics of a complex nonlinear system.

Cellular automata give us a new perspective on the origin of life. Von Neumann's self-replicating automaton is so special that it is carefully designed to replicate a highly complex initial configuration. In 1993, researchers developed a cellular automaton with 29 states, in which randomly selected initial states resulted in more than 98% of self-replicating structures. In this automaton, the self-replicating entity is almost certain.

Complex systems support the idea that on an inanimate planet with sufficiently complex chemical composition, life is likely to form spontaneously and organize into more complex forms. What remains to be understood is what laws in our universe lead to the spontaneous emergence of self-replicating structures, that is, what laws of physics make this critical first step to life not only possible, but also inevitable.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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