A little bit about chaos and how to create it

Speaking of "chaos", we usually mean a complete lack of order, absolute disorder and randomness. From a mathematical point of view, chaos and order are not mutually exclusive concepts. Chaos theory (there is something fascinating in the names of mathematical theories) is a fairly young mathematical field, the creation of which is equated by the significance of the discoveries of the twentieth century to the creation of quantum mechanics. Chaos happens in nonlinear dynamic systems. In other words, any process that takes place over time can be chaotic (for example, tree height, body temperature, or the population of Madagascar cockroaches).

To understand what chaos is, we first turn to systems that are not endowed with such a feature. Deterministic systems do not allow any accidents: the value at the output is completely determined by the values ​​at the input. Thus, a change in the initial conditions causes a proportional change in the result. Thus, Newtonian mechanics imply determinism, and by changing, for example, the force of a kick on the ball, we can expect a corresponding change in the duration of the flight of this ball. So, according to the principle of determinism, the position of the ball is currently completely determined by the position of the ball at the previous moment and the future position depends on the current one and it is not difficult to calculate all this. Thus, past astronomers fully trusted this principle and believed that the universe is a strictly deterministic system and the position of celestial bodies in the future (and in the past) can be calculated knowing their current position and speed, i.e. knowing the initial conditions. It was assumed that the more precisely the initial conditions are known, the more accurate the forecast result will be, however, the famous mathematician Henri Poincaré, who (in his spare time probably engaged in describing the orbits of celestial bodies), found that in systems of 3 or more bodies, with a slight change of initial conditions (position and speed), the trajectories of the body very quickly move away from each other. Two close sets of initial conditions gave different results.

Meteorologist Edward Lorenz made a great contribution to the theory of chaos. In the sixties of the last century, this American worked on a computer program simulating the movement of air masses in the atmosphere of the Earth. We all know that a computer (despite popular rumors) is a strictly deterministic system, and this creates the well-known principle of “garbage in garbage out”. Lorenz drove his program to the tail and mane, getting all sorts of different results. Some of his colleagues even suggested that this model was an accurate predictor of the weather, asked if they would take an umbrella tomorrow. Of course, these conclusions were hasty, soon one feature of the weather model became clear. Once to speed up the calculations, Lorenz launched the program not first, but entered into it the data from the previous “run”, which was printed on paper. However, the results of such a launch quickly began to deviate from those already received, forming a completely different picture. A bit unexpected, right? It turned out that Lorentz did not introduce the exact results of past calculations, but rather rounded up before printing, this error was simply ignored. The Lorentz model turned out to be hypersensitive to initial conditions. The slightest difference in the input data led to a strong discrepancy between the results over time. This dependence on the initial conditions and was called chaos. Lorentz voiced the famous feature of chaos, called the "butterfly effect", which suggests that depending on whether a butterfly flaps its wings in the forests of Brazil, it depends on whether a hurricane happens in Texas or not. The same principle was used as the basis for the film with Ashton Kutcher of the same name (unscientific cinema, not necessarily to watch).

Deviation in the results of repeated calculations

All this dependence on initial conditions suggests that we cannot make long-term forecasts in unstable dynamic systems. Any error in the initial conditions will not allow us to predict the result for any long period of time. If, for example, we take the Lorentz model, as input for determining wind speed, we will need to enter temperature and pressure values ​​at each point of the earth’s atmosphere, only then we can expect a reliable forecast for a long period. Moreover, the input data must be absolutely accurate, that is, with an infinite number of decimal places. And as you know, absolutely all measuring instruments on Earth have a non-zero error. No matter how accurately the value is measured, it is always possible (theoretically) to measure more accurately. And there are no such machines that would allow to enter an infinite number of decimal places. Maybe with the advent of quantum computers, something will change, I do not know.
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So it turns out that there is no way out of chaos and we must put up with it. But not everything is so bad, in my opinion. If all processes in the universe would be completely deterministic, without a single hint of chance, it would be much more boring to live. Some scientists are even inclined to think that chaos gives the universe an "arrow of time", a directional and irreversible movement from the past to the future.

However, “chaos” and “randomness” are not equivalent. A definite interpretation of seemingly random processes puts them in order. For example, the time between the beats of a person’s heart is variable, even if a person is not subject to physical stress. If we observe the heartbeat for some time and the intervals between the beats are written in the table, we also create the second column by copying the values ​​from the first, but shifted by one value (i.e. the first dimension (t) in the first column will correspond to the second dimension (t + 1) in the second, the second - the third, and so on;), it will be possible to build a map, where we will have vertical values ​​without shift (t), and horizontally - values ​​with shift (t + 1). The points on this map will not be scattered in random order, but will be drawn to a certain area, forming an attractor.

A common example of a chaotic system is a double pendulum, i.e. the pendulum, to the end of which is attached the second pendulum. You may have seen similar pendulums in gift shops. So if you take two identical pendulums, put them side by side and reject them approximately by an equal amount, then after a few oscillations the pendulums will completely become out of sync. The more precisely we will observe the initial conditions, the longer the pendulums will swing in time, but there is no escape from the divergence.

Such patterns draws a light bulb on a double pendulum artist George Ioannidis

For a long time, chaos theory was considered a kind of mathematical abstraction that does not have confirmation in real conditions. This problem worried a Japanese man named Leon Chua, who was aiming to show that chaos could be created. For this purpose he assembled an electrical circuit.

Chua chain was the first electrical circuit capable of generating chaotic signals. His creation was ingenious in its simplicity, the circuit consisted of four linear elements: two capacitors, one inductance and a resistor, and also included one nonlinear locally active element, on the piecewise linear volt-ampere characteristic of which there was an area with negative resistance. This element is now often called the Chua Diode. The circuit is a generator, and the Chua diode is a necessary part for achieving chaotic oscillations. This element is not available as a separate component, but it is easy to assemble it using two operational amplifiers. Other ways of implementing this nonlinearity include a pair of inverters connected in parallel or a tunnel diode (it looks like it is still available as a separate component), which, as is known, has a “valley” on the IVC.

Generalized Chua generator circuit and equations describing it

The math behind all this is quite complicated, but if you don’t go into the jungle, this circuit is described by three differential equations showing the time variation of the voltage on the two capacitors and the current through the inductance. Numerical solution of these equations shows that with certain ratios between the components of the chain, the change in the values ​​of variables over time becomes chaotic.

Collect the generator Chua work is not special. This chain can demonstrate such phenomena of chaos as bifurcations and chaotic attractor. However, to observe all these wonders, an oscilloscope will be needed, and even with two inputs. In the classic version, the circuit consists of two capacitors, one inductance, seven resistors, a chip with a pair of operational amplifiers and two 9V batteries (you can use a power supply, but the power supply should be bipolar). In order to achieve chaotic behavior, certain ratios should be observed between the values ​​of the elements. So, the capacitance of the capacitor C2 should be approximately 10 capacitances C1, the ratio C2 / C1 is called α. The coefficient β indicates the relationship between R, C2 and L, namely, β = R ^ 2 C2 / L and should be approximately 15.

Schematic diagram of the generator with negative resistance on the operational amplifiers

So, proceed to the assembly. You can also assemble on the breadboard, but in order for the signals to be clearer, it is better to solder the components on the printed circuit board. In my assembly I used 47nF and 470nF capacitors, an inductance of 15mH and a potentiometer of 1 kΩ (for lack of one with a nominal 2 kΩ, connected it in series with a resistor of 1 kΩ). Consistently with inductance it is possible (but not necessary) to include a resistor of small nominal (up to 10 Ohm) to add "beauty" to the signals. Chua's diode is implemented in the standard way using two opamps. I used the TL082CP chip, according to the specification, this is a broadband operational amplifier, I advise you to use this type, with simpler analogs, I didn’t have a circuit. To create a characteristic with the necessary slopes, we need the following resistor ratings: R1 = R2 = 220 Ohm, R3 = 2.2 kΩ, R4 = R5 = 22 kΩ, R6 = 3.3 kΩ. The opamp can be powered by two 9V batteries, we need bipolar power for the OS to work correctly. My assembly is clumsy, I agree - the wiring for power and twisted resistors, other minor flaws, but this was enough to monitor the chaotic signals.

We will save the rest of the fee for the following projects

After a neat assembly of this simple scheme, you can try to see what kind of signals it generates. The signals will be removed from the capacitors C1 and C2. On my circuit, I made two BNC connectors for easy connection of the circuit to the oscilloscope. We connect the cables to the oscilloscope and select the XY mode, when along the same axis we have the voltage on the first capacitor, and on the other, the voltage on the second. What to display on X, and what on Y does not matter. Unscrew the potentiometer knob to the maximum value and power the circuit. A dot should appear on the oscilloscope screen. Slowly decrease the resistance value (it is better to use potentiometers with a large stroke and with a large knob in order to ensure a smooth change in resistance), at some point the point should turn into an orbit. The subsequent decrease in resistance leads to a doubling of this orbit, we begin to observe bifurcations. Doubling the period of the orbit will continue to occur with decreasing resistance, the distances between the subsequent bifurcations will constantly and systematically decrease. Those. the difference in resistance between the quadruple and the eight orbit will be less than between the quadruple and double. The rate at which the interval between bifurcations decreases is determined by the Feigenbaum constant. The period to which you can observe bifurcations depends on the clarity of the signals (that is, on the quality of the connections) and on the sensitivity of the potentiometer (hand trembling is also not good). At some point, a stable orbit gives way to a two-loop attractor, which marks the onset of chaos. This attractor has three equilibrium points: one at the origin of coordinates, and two in the "holes" of loops. A typical trajectory of an attractor begins to rotate around one of the “holes”, moving away from the equilibrium point with each turn, then the trajectory either returns closer to the center and again moves away, or goes to another equilibrium point, where the process repeats. The number of rotations in each case is random.


Chaos formation through bifurcations

This attractor will exist in a certain resistance range, and then give way to a stable orbit, showing harmonic oscillations. At sufficiently small values ​​of resistance, the circuit turns into a simple oscillating circuit, generating a sinusoidal signal with a frequency determined by the values ​​of capacitors and inductance. For greater "flexibility" of the circuit, the potentiometers can be replaced by resistors in the negative resistance circuit.

If we look at the spectrum of signals, we see that in the chaotic mode, the generation band is quite wide and does not have pronounced peaks, besides it starts with a constant component.

Chaotic signal spectrum

The scheme is extremely simple, but its behavior has been studied by many scientists working with the theory of chaos. With its help, bifurcations were studied and a whole gallery of various attractors was created. However, apart from purely scientific interest, this scheme has a practical application.

Since this is a generator, it means that it can be used for radio communication, and since this generator is unusual, radio communication can be made secure. There are several types of modulation of a chaotic signal, from simple masking of an information signal to high-level digital modulation. The high sensitivity of a chaotic oscillator allows it to be used as a detector of weak signals. Also reported on the creation of a random number generator based on this scheme. In addition, as you noticed, the spectrum of this generator lies in the sound range, so conceptual musicians did not fail to take advantage of this scheme.

I don’t know how many people will want to assemble this chaotic generator, because it’s not enough practical use, but I think the opportunity to play around with it and watch interesting patterns on an oscilloscope is worth these penny details and half an hour. Even if you buy all the components by the piece in the store, 200 rubles is the maximum that you can spend, but I am sure that many people have all the details in the store!

This scheme may be interesting to students of mathematical and electrical engineering departments. I think that the demonstration of the work of the Chua generator will be able to interest teachers, whose scientific interests include chaos theory. Thank you all for your attention!