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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), the original title: "build a Tsai's circuit at the weekend to experience the 'fox essence' (tinker salon). Author: Wu Jianyong (Professor, Department of Neuroscience, Georgetown University, USA)

Tsai's circuit, there are "foxes" in the circuit, charming and charming, only because of chaos. When it comes to chaos, many people may only know the butterfly effect. Chaos is everywhere, and it is not surprising that chaos occurs in the circuit. With a little learning about electronic circuits, you can build a Chua's circuit yourself.

The circuit is like an article.

A circuit is an article written with electronic components. Electronic components are like Chinese characters, which are connected together into sentences and then into articles. There are only a few dozen words in each of the three hundred Tang poems, and there are only a few basic components with tens of millions of circuits. Of course, there are thousands of circuits in the world, so why do we have to build a "Tsai's" circuit? Probably because ordinary circuits are like eight-part essay, although the beginning and turning point is meticulous, but the reader can see through at a glance, it is all boring. And Tsai's circuit is like the story of Liaozhai, which contains foxes, charming and charming, sometimes butterflies, sometimes stormy, cloudy and sunny, and there is not a trace of repetition. It has attracted veteran drivers all over the world like vultures, with tens of thousands of research articles and water in various scientific journals. Although I have a biological dog, I can't be seduced, so I get involved on weekends.

How can a fox be called Qianjiaobai? It's just that electrons don't run regularly in the circuit. We all know that there are a lot of electrons running in the circuit. A small circuit also has 100 billion electrons. In general circuits, electronics follow the rules, one way to work, one way home, you can see retirement at a glance. In Tsai's circuit, the electrons are restless, they can run one pole at a time, and they can move forward and backward for a while, but the directional flow is unexpectedly variable. The most powerful supercomputer in the world can only estimate the future, but can never accurately predict the future.

It's a bit like the weather forecast, wind, cloud, rain and snow, the supercomputer can only estimate it, but it won't work for more than a week. The current in Chua's circuit is similar to the turbulence in the atmosphere, which is the so-called "chaotic" behavior.

Turbulence can be seen everywhere, the whirlpool in the river and the elegance in cigarettes can be seen everywhere. But after all, these happen in the wild, strange, fleeting and difficult to study. The scientific gap that Tsai's circuit needs to fill is to build an everlasting whirlpool in the laboratory so that you can see enough. On the screen below is the whirlpool generated by Tsai's circuit.

The origin of Chua's circuit from https://www.chaotic-circuits.com/8-simulating-chaus-circuit-with-ltspice/ is inspired by chaos warrior Lorentz. When Lorentz used computers to study weather forecasts, he found that using the same data, the forecasts calculated by small computers and large computers were completely different. This is because small computers are accurate to three decimal places (sorry, 1960s), while large computers can be accurate to six digits. So is it more accurate when the computer is bigger? Ordinary people may think so, but Edward N. Lorenz is not an ordinary person. The mathematician who used to be called "Quan Xue er" in the old Beijing dialect actually saw that it was not the problem of computer accuracy, but the unpredictability of the problem itself. It is unpredictable because there is chaos in atmospheric turbulence, that is, the "butterfly effect" of the street has been smelled by the popularity of Chinese science. Lorentz expressed his ideas in mathematical language and obtained the Lorentz equation (figure 1).

Fig. 1 the Lorentz equation (left) and the curve drawn with the equation (right) do look like a butterfly. Look at the butterfly running out of the Lorentz equation. Is there a resemblance to the curve in the oscilloscope in the picture? Chua's circuit uses electronic components to realize the essence of Lorentz equation. His inventor, Leon O. Chua,1936-, a Chinese scientist, is as smart as Lorentz. Lorenz described this unspeakable secret of nature in mathematical language, while Cai described the same idea in the language of electronic engineering.

There is another story about the invention of Chua's circuit. After the Lorentz formula was published, Professor Matsumoto of Waseda University in Japan was determined to use electronic components to build the Lorentz equation. It is a pity that he has been busy for three years, but he is on the wrong track. Because he uses a circuit to simulate every parameter in the Lorentz equation, as a result, the circuit is getting bigger and bigger, and thousands of elements cover a large table, but it can only be similar in shape and can not produce chaos. Mr. Cai was originally a visiting scholar who came to study in Matsumoto Lab. On the first day he came to the laboratory, he glanced at the big table and was almost stimulated, thinking that the big boss must have been rolled in. Mr. Cai thought that the Lorentz formula has only two unstable points, which may be realized by only a few nonlinear elements. When he returned to the hostel, he couldn't sleep at night, looked everywhere for paper, and finally drew a simple circuit with only five components on the tissue. The next day he gave it to the big boss, and sure enough, chaos could be generated when he tried. One day defeats three years, and five components beat a big table. I have always remembered the story of warm wine defeating Hua Xiong, but the success of scientists is nothing more than this. Cai Shaotang is the father of a "tiger mother". Tiger mother-style training pays attention to rigid repetition and fatigue bombing. On the other hand, Mr. Cai is a model of empathy, and the contrast is great.

Fig. 2 basic Chua's circuit. L, inductor coil; C1, C2, capacitor; R, resistor; NR, Chua's diode. For the functions and characteristics of each component, see the text. Let me show you about Tsai's circuit. Don't be afraid. It doesn't matter if you don't understand electronic engineering. I haven't learned it either. Electronic engineering is just a specialized language like a foreign language. Although the language does not understand, the truth is still very easy to understand, as long as the terminology is translated into everyday language, you can also understand the wonder of the story.

There are only five components in Tsai's circuit, the inductor L, the capacitor C1 and C2, the resistor R, and a strange NR. The common feature of all components is that they all have two "legs" up and down, and the power supply flows in and out. For example, the current comes in from above, goes out from below, or goes out from below. The horizontal black line in figure 2 is the wire, and the black dot is the contact between the wire and the component. So another common feature is that the legs of all components are connected to two separate wires, so that the current flowing from one component can follow the wire into the other.

To understand the mysteries of all circuits, you have to start with two leg elements connected together. To clarify, let's first study a more famous circuit (figure 3), which is the flashlight circuit that illuminates the world.

Figure 3 A simple circuit connected by a battery and a light bulb in the circuit in figure 3, the battery (left) and the light bulb (right) are two-legged electronic components, connected by wires (red lines in the picture). A wonderful phenomenon appeared: a small light bulb shines brightly! Specifically, the chemical energy in the battery drives a large number of electrons to flow through the light bulb, and the electrons collide with the metal lattice in the filament to release energy, heating the filament and glowing. In such a simple circuit, about 30 billion electrons flow through every second. As soon as the wire was cut off, the current stopped and the light bulb went off.

The famous principle in electronics, the flashlight circuit is simple, but it contains one of the most important principles in electronics, that is, the circuit has to go back and forth, and the current flowing through the circuit is exactly the same. In figure 3, the upper and lower green arrows indicate the direction of the current, which flows to the light bulb above and out below. One round must be complete, and the current value back and forth is exactly the same. You see, such a few vernacular words clarify Kirchhoff's law that you spent a lot of tuition to learn in college. The tutor who taught me the circuit was Li Lianzi, who repaired mobile phones at the intersection. He famously said, "if you learn Kirchhoff's law, you will not be afraid to travel all over the world." When you see any electrical appliance again, you have to subconsciously look for its legs, where the current comes in and where it comes out.

A friend who likes to cheat said, Why does my iPhone headset have only one cable? This is for convenience by making the two threads into one strand. Why don't you cut the wire? There is also a friend who likes to cheat and say that the computer chip has hundreds of legs. How can the road count this time? In fact, the chip is just a combination of many two-legged components. For example, the way the current comes back can be shared, which is called a ground wire (the line below in figure 3 can be called a ground wire), and the other hundreds of wires represent hundreds of batteries or two-leg components like a light bulb.

If you can read this patiently, you are already in the middle of the teacher. The following analysis of the fox in Tsai's circuit, we need to have some fox mentality. Of course, the method is still the same, first consider the two connected components in figure 4A. The bending element on the left is called inductor (L), and the chip on the right is called capacitor (C).

Figure 4 LC circuit A: a LC circuit composed of a coil and a capacitor, with the circuit diagram in the middle, the real coil on the left and the capacitor on the right; the behavior of the B:LC circuit: the current flows back and forth, or "oscillation"; C: because of the energy loss in the circuit, the amplitude of the oscillation will be lower and lower. It doesn't matter if you don't understand inductance and capacitance. You must have been in love, haven't you? Must know that a successful relationship must be half-pushed. Inductors and capacitors are connected together like a pair of half-and-half lovers. The capacitor is like the active side, a current, hot over, it does not dodge at all, according to the order to receive all. On the other hand, the inductor is like the passive party, which begins to resist completely when the other party discharges, and then slowly accepts it. The two are a perfect match: at first, the electrons are received by the capacitor, so the current flows into the capacitor; then the inductor gets off the shelf, and the current flows back and forth from the capacitor to the inductor. Like the left side of figure 4B, the current flows from the inductor to the capacitor and then from the capacitor to the inductor. This kind of back and forth is called "oscillation" in electronic engineering jargon. Of course, oscillations, like falling in love, can't be sweet forever. So the actual situation is like figure 4C, over time, the passion will be less and less, and the amplitude of oscillation will be lower and lower.

Tsai's secondary tube usually needs a billionaire's father-in-law or father-in-law to provide continuous financial support for the protagonist in order to make the love story fluctuate. The same is true of the oscillating circuit, which needs to be constantly supplied with energy in order not to die out like the curve in figure 4C.

The "Chua's diode" in Tsai's circuit (the rightmost NR in figure 2) is such a "paying role". There are many ways to provide energy to the oscillation circuit, and this Chua's diode is a strange work, with which it can produce strange chaos. Why is it strange? Because it has an abnormal characteristic called "negative resistance". We know that in a normal two-legged element, the higher the voltage at both ends, the greater the current flowing through the element. On the other hand, the Chua's diode is just the opposite. When the voltage at both ends decreases, the current flowing through it increases. In this way, as long as the amplitude (voltage) of the oscillation is low, the Chua's diode will provide additional current to allow the oscillation to continue.

Tsai's diode is a non-existent component, which is imagined by Mr. Cai. According to Mr. Cai, his original idea was to satisfy the conditions of the Lorentz equations with the least number of elements. So he makes multiple sketches and then uses Kirchhoff's law to merge the components together (the process is a bit like the formula we do when we do math problems). In the end, there are only five components left in the circuit, but one of them is the "negative resistance" of the two leg elements that do not exist in nature. Although this negative resistance does not exist in nature, it can be defined by a formula. So the circuit that Mr. Cai gave to Professor Matsumoto the next day was successfully tested on a circuit simulator on the computer. The real circuit was made by Mr. Cai's students half a year later. The circuit we are building now uses two operational amplifiers to simulate Chua's diodes (figure 5). However, only the wonderful work of Tsai's diode is not enough to produce chaos. Now I'll give you a surprise gift bag.

What happened? If the mistress takes a closer look at the Chua's circuit in figure 2, there will definitely be a big discovery that there are two capacitors. Except for the normal romantic couple composed of inductor and capacitor on the left, the capacitor (C1) in the middle is clearly a third party who plays an active role. It is not wrong to say that it is a third party, because there is a resistance between it and the normal couple on the left (inductance and capacitance). I don't have to say that you can understand that all mistress stories that are involved in love should have all kinds of isolation, and the isolation should not be too small or too big. The isolation is too big and the mistress can only be ignored like the air, while if the isolation is too small, it will lead to a great tragedy, which will lead to fragments in one episode of the TV series. So medium-sized isolation is the key to producing foxes.

To be honest, this Chua's circuit is much better than a dog-blooded TV series, which uses this mistress to provide the two unstable states described in Lorentz's formula. The idea is as follows: when a LC circuit and a Chua's diode are used to form the circuit, there will be oscillation, which is an unsteady phenomenon of electrons running back and forth; and if a mistress is added and a new unstable factor is introduced, chaos can be generated. According to this routine, many scholars have studied Chua's circuit systematically, and basically think that there must be a half-push oscillator in the circuit that can produce chaos. Interestingly, this is still an unproved conjecture called Elwakil and Kennedy conjecture [2].

Build a circuit as simple as Tsai's circuit, and it's so magical that people with obsessive-compulsive disorder like me will simply waste their lives if they don't play with it. It is worth comforting that there are many strange people like me in the United States. When I look it up on the Internet, I share a lot of experiences. So I built one without much effort (figure 5). As mentioned earlier, Tsai's diode is an imaginary component that does not actually exist. But many people on the Internet, according to Mr. Cai's idea, can be built with two operational amplifiers and a few resistors (figure 5B). One of the key elements in the circuit is the resistor, which adjusts the degree of coupling between two nonlinear elements. As mentioned above, if the coupling degree is small, the circuit will not produce chaotic attractors (the influence of the mistress is regarded as air), while when the coupling degree is too high, the line loses oscillation and is stabilized by Chua's diodes at high or low levels.

Fig. 5 the Chua's circuit I built. An is the actual component on the circuit board. Two operational amplifier chips (an enamelled coil wound around a ferrite core) and several resistor and capacitor elements can be seen in the picture. The operational amplifier is powered by a 9v battery to realize the function of Chua's diode. B is the circuit diagram, which marks the actual resistance and capacitance value. Interested netizens can click on this picture. Any type of operational amplifier is fine. I used the LF356 I already have. Chaotic attractor here I'm going to teach you a new word called "chaotic attractor". What Chaos is all about is still God's secret, and human beings still don't understand it. But because chaos is widespread, it is talked about by everyone. When talking to others about chaos, if you only know the butterfly effect, you will be seen by others as very low and greasy. If you say "chaos attractor", the B box will be much higher, not only ordinary people do not understand, but also can bluff when you run into an expert.

The concept of attractor is easy to understand, which is the fact that a moving object is regulated on a set of orbits. Swings, for example, always run in an invisible space orbit. If the swing is pushed, it may be crooked from side to side, but it will eventually return to that track. This characteristic of coming back on its own even if pushed away is a bit like being attracted, so this state is called attractor. Note that the attractor is a "state" rather than a physical object. For example, people on the playground are willing to run along the track. This phenomenon can be called "runway attractor". Note that the real thing of the runway is not the attractor, but the phenomenon that the attractor runs along the track.

Back to chaos, the butterfly-shaped orbit (right of figure 1) must also be an attractor, but because it has two wings and two sets of attractive orbits, it is also called a "strange attractor" (strange attractor). Of course, strange attractors are not chaotic, so the butterfly orbit drawn by Lorentz formula is also called "Lorentz strange attractor".

It's really cool to use your own Chua's circuit to pull out all kinds of attractors. Figure 6 shows the different attractors formed when the resistance values in the circuit are different. As mentioned earlier, when this resistance changes, Chua's diodes and capacitors have varying degrees of influence on LC circuits: when Chua's diodes and capacitors have the least impact on LC circuits, LC circuits appear regular oscillations, or ordinary attractors (left of figure 6); when the influence increases, strange attractors gradually appear on the line (figure 6); and when the influence reaches moderate, chaos occurs, that is, stable Lorentz attractors.

Fig. 6 the dynamic characteristics produced by Tsai's circuit have been criticized by my classmates who have studied physics, saying that I have been talking about chaotic attractors for a long time without listing a single formula. To put it politely, it is not strict, but to put it bluntly, it is a civil science. But I believe what Master Li said, "people who list formulas all the time can't actually use 'human words' to make sense." Although the formula is also a kind of human language, few people understand it. Therefore, if you can explain the scientific truth clearly in common words, try not to use formulas, and avoid situations in which the truth is unclear although formulas can be listed (such as quantum mechanics).

The use of chaotic circuits has been talking about chaotic attractors for a long time, but what is the use except freshness? It's of great use. The most secure communication, for example, requires a character and a password and is never reused. But where did the password come from? This requires a random number generator. In fact, the random number generator in the computer is not really random, and the pattern can still be found if it is used too much. The chaotic circuit is a real random number generator [3]. A similar use is for robots to find a dead end exit algorithm, that is, first walk around, and then carefully analyze [4]. Another magical use is called 'chaotic synchronizer', which is that two chaotic circuits with similar structures can be coupled so that for both sides of the coupler, the paths are exactly the same and know where they are, while to other observers, the path looks like an unexplained chaotic state [5].

So far, many scientists have been studying how to simplify Chua's circuit, such as using a real two-leg element (memristor, memristor) to replace the Chua's diode built by multiple elements. The purpose of simplifying Chua's circuit is to build tens of thousands of chaotic circuits on a small chip, which may be able to simulate some characteristics of the brain's neural circuits.

What interests me most as a neuroscientist is the relationship between chaos and the generation of thinking. We know that every brain cell is an oscillator, and each brain cell is generally connected to thousands of other brain cells. do so many coupled oscillators have a lot of chaotic behavior? The answer is yes. But do these complex chaotic behaviors have anything to do with the generation of thinking? Furthermore, does the generation of thinking depend on the line of chaotic behavior? These questions have aroused the interest of a generation of scientists, but so far they have not been solved. The interconnected network of brain cells is by far the most complex network in the universe. This network must have many unique properties. I'll leave it to you to make up for it, brother. If I say one more word, I will expose my ignorance.

Conclusion for those of you who are interested in chaos science, you should accept my brother's lesson that to learn science, you must first learn not to speak human language (that is, you can use mathematical tools). Many concepts need mathematical tools to understand, such as the Hauszov dimension (Hausdorff-Besicovitch dimension [6]), which was originally used to describe smoothness, but is now used to describe chaotic typing [6]. Using Haussov dimension, chaotic typing is mostly non-integer or even irrational. For example, the surface of cauliflower can be drawn with 2.7 dimension, and the alveoli can be drawn with 2.97 dimension. I'll send you here first today.

Colored eggs-- you will be an expert if you understand some of the concepts of dynamic systems.

Anosov diffeomorphism

Arnold tong

Axiom A system

Bifurcation diagram

Box-counting dimension

Correlation dimension

Conservative system

Ergodicity false nearest neighbors

Hausdorff-Besicovitch dimension

Invariant measure

Lyapunov stability

Measure-preserving dynamical systems

Mixing Poinka é section

Recurrence plot

SRB measure

Stable manifold

Topological conjugacy

reference

[1] "Science Popularization Tiger Mom and his Dad" was written by me in 2012 and published in the print magazine Physics. Later, Wechat was also forwarded from the media, and there are a lot of copywriters. Search it with the name of the article.

[2] A. S. Elwakil and M. P. Kennedy. (2000) Chua's circuit decomposition: a systematic design approach for chaotic oscillators. Journal of the Franklin Institute 337 (2000) 251 purl 265.

[3] Bonilla, L.L., Alvaro, M. & Carretero, M. Chaos-based true random number generators. J.Math.Industry 7, 1 (2016). Https://doi.org/10.1186/s13362-016-0026-4

[4] Ch.K. Volos, I.M. Kyprianidis, I.N. Stouboulos. (2012) A chaotic path planning generator for autonomous mobile robots. Robotics and Autonomous Systems Volume 60. Issue 4 2012.

[5] 5 Kinzel W., Englert A. and Kanter I. (2010) On chaos synchronization and secure communication. Phil. Trans. R. Soc. A.368379-389

[6] https://en.wikipedia.org/wiki/Hausdorff_dimension

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