It may seem to you that Chaos Theory is very far from the stock market and trading in particular. And indeed, how can one of the branches of mathematics, which deals with complex dynamic systems of a nonlinear nature, relate to the world of trading? But it can!
The peculiarity of nonlinear systems is that their behavior is directly dependent on the initial conditions. But even specific models do not allow us to predict their future behavior.
There are many examples of such systems on the planet - turbulence, atmosphere, biological populations, etc.
But, despite their unpredictability, dynamic systems strictly obey one law and can be simulated if desired. For example, in the stock market, traders and investors also encounter curves that can be analyzed.
A little history
Chaos theory found its application back in the 19th century, but these were only the first steps. Edward Lawrence and Benoit Mandelbrot began studying this theory more seriously, but this happened later - in the second half of the 20th century. At the same time, Lawrence tried to predict the weather in his theory. And he managed to deduce the main reason for its chaotic behavior - different initial conditions.
Basic Tools 
The main tools of Chaos theory include fractals and attractors. What is the essence of each of them? An attractor is what the system is attracted to and where it ultimately tries to arrive. Its value is most often a statistical measure of chaos as a whole. In turn, a fractal is a kind of geometric figure, part of which is constantly repeated. By the way, it was on this basis that one of the main properties of this instrument was derived - self-similarity. But there is one more property - fractionality, which becomes a mathematical reflection of the degree of irregularity of the fractal.
At its core, this tool represents the opposite of chaos.
Unfortunately, there is no exact mathematical system of Chaos theory for studying market prices. Therefore, there is no need to rush to apply the theory of Chaos in practice. On the other hand, this direction is one of the most popular and worthy of attention.
Chaotic markets
As practice shows, most modern markets are subject to certain trends. What does it mean? If you look at a curve over a long period of time, you can always see the reason for a particular movement. But not everything is so smooth. There is always a certain element of unpredictability in the market, which can be introduced by some kind of catastrophe, political events or the actions of insiders. At the same time, modern Chaos theory tries to predict changes in the market taking into account some neural network approaches.
Possibility of system modeling
Experienced participants know very well that it operates on the basis of some complex system. This is not surprising, because there are many participants in it (investors, sellers, speculators, buyers, arbitrageurs, hedgers, and so on), each of whom performs their own tasks. Moreover, some models describe this system, for example, Elliott waves.
Difference between the Mandelbrot distribution and the normal distribution
In practice, the price distribution has a much wider spread than most market participants expect. Mandelbrot believed that price fluctuations have infinite variance. This is why any analysis methods are ineffective. They were asked to analyze the price distribution solely on the basis of fractal analysis, which showed itself to be the best.
conclusions
Bill Villas (author of the book “Trading Chaos”) is confident that the characterizing links of chaos are systematicity and randomness. In his opinion, chaos is permanent, in comparison with the same stability, which is temporary. In turn, this is a product of chaos. In essence, Chaos Theory questions the very basis of technical analysis.
According to Williams, a market participant who only takes a linear perspective in his analysis will never achieve great results.
Moreover, traders lose because they rely on various types of analysis, which are often completely useless.
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Chaos theory, also known as the theory of nonlinear dynamic systems, has recently become one of the most fashionable approaches to market research. Unfortunately, an exact mathematical definition of the concept of chaos does not yet exist. Now chaos is often defined as the extreme unpredictability of constant nonlinear and irregular complex motion that occurs in a dynamic system.
According to chaos theory, randomness and order reign together in the world. They are inseparable, like good and evil, like left and right.
It should be noted that chaos is not random, despite the property of unpredictability. Moreover, chaos is dynamically determined (determined). At first glance, unpredictability borders on randomness - after all, we, as a rule, cannot predict just random phenomena. And if we treat the market as a random walk, then this is the same case. However, chaos is not accidental; it obeys its own laws. According to chaos theory, if you talk about chaotic price movement, then you should not mean a random price movement, but another, especially ordered movement. If market dynamics are chaotic, they are not random, although they are still unpredictable.
The unpredictability of chaos is explained mainly by its significant dependence on the initial conditions. This dependence indicates that even the smallest errors in measuring the parameters of the object under study can lead to completely incorrect predictions. These errors can also arise due to elementary ignorance of all initial conditions. Something will definitely escape our attention, which means that already in the very formulation of the problem there will be an internal error, which will lead to significant errors in predictions. In relation to the inability to make long-term weather forecasts, the significant dependence on initial conditions is sometimes called the “butterfly effect.” The "butterfly effect" refers to the possibility that the flapping of a butterfly's wing in Brazil will result in a tornado in Texas.
Additional inaccuracies in the results of research and calculations can be introduced by the most invisible factors influencing the system at first glance, which appear during the period of its existence from the initial moment until the appearance of the actual and final result. In this case, influencing factors can be both exogenous and endogenous.
A striking example of chaotic behavior is the movement of a billiard ball. If you have ever played billiards, then you know that the final result depends on the initial accuracy of the shot, its force, the position of the cue relative to the ball, the assessment of the location of the ball being hit, as well as the location of other balls on the table. The slightest inaccuracy in one of these factors leads to the most unpredictable consequences - the ball may roll in a completely different direction than the player planned. Moreover, even if the player did everything correctly and fantastically successfully, try to predict the movements of the ball after five or six collisions.
Let's consider another example of the influence of initial conditions on the final result. Let's imagine, for example, a stone on top of a mountain. Just push it a little and it will roll down. However, a very small change in the strength of the push and its direction can lead to a very significant change in where the stone stops at the foot of the mountain. There is, however, one very significant difference between the movement of a stone and the behavior of a chaotic system. In the first, the factors influencing the stone during a fall from a mountain (wind, obstacles, changes in the internal structure due to collisions, etc.) do not have a strong influence on the final result compared to the initial conditions. In chaotic systems, small changes have a significant impact on the outcome not only in the initial conditions, but also at any other point in time.
One of the main conclusions of chaos theory, therefore, is the following: it is impossible to predict the future, since there will always be measurement errors, generated, among other things, by ignorance of all factors and conditions.
In other words, small changes and/or mistakes have big consequences.
No matter how much you start counting, the result will almost always be different (Fig. 6.3). Moreover, the coincidence of results will occur less frequently the further into the future we look. This does not refer to exact mathematical formulas, but reflects the life paradigm of chaos theory. There is a good proverb that characterizes this formulation of a theoretical postulate: “You cannot step into the same river twice.”
Another basic property of chaos is the exponential accumulation of error. According to the laws of quantum mechanics, the initial conditions are always uncertain, and according to chaos theory, these uncertainties will quickly grow and exceed the permissible limits of predictability.
tttt
Therefore, I cannot accurately predict the future, but only guess
I don't know all the initial conditions
I don't know all the influencing factors
Time >
T(0)
T(p)
T(1)...T(p-1)
Rice. 6.3. Significant dependence of the result on the initial conditions and influencing factors.
The second conclusion of chaos theory: the reliability of forecasts quickly decreases over time (Fig. 6.4). This conclusion is a significant limitation for the applicability of fundamental analysis, which, as a rule, operates with long-term categories.
Time
They usually say that chaos is a higher form of order, but it is more correct to consider chaos as a completely different form of order - inevitably in any dynamic system, order (in its usual understanding) is followed by chaos, and chaos is followed by order. If we define chaos as disorder, then in such disorder we will definitely be able to see our own special form of order. For example, smoke from a cigarette, which initially rises in the form of an orderly column, under the influence of the external environment takes on more and more bizarre shapes, and its movements become chaotic. Another example of randomness in nature is a leaf from any tree. It can be argued that you will find many similar leaves, for example oak, but not a single pair of the same
leaves. The difference is predetermined by temperature, wind, humidity and other external factors, as well as purely internal reasons (such as, for example, genetic differences).
Movement from order to chaos and back, apparently, is the essence of the Universe, no matter what manifestations of it we study. Even in the human brain there is both order and chaos at the same time. The first corresponds to the left hemisphere of the brain, and the second to the right. The left hemisphere is responsible for conscious human behavior, for the development of linear rules and strategies in behavior, where “if... then...” is clearly defined. In the right hemisphere, nonlinearity and chaos reign. Intuition is one of the manifestations of the right hemisphere of the brain.
Chaos theory studies the order of a chaotic system, which appears random, disordered. At the same time, chaos theory helps to build a model of such a system, without setting the task of accurately predicting the behavior of a chaotic system in the future.
The first elements of chaos theory appeared in the 19th century, but this theory received true scientific development in the second half of the 20th century, together with the work of Edward Lorenz from the Massachusetts Institute of Technology and the French-American mathematician Benoit B. Mandelbrot ( Benoit V. Mandelbrot).
Edward Lorenz at one time (early 60s of the 20th century, work published in 1963) considered the difficulties in weather forecasting.
Before Lorenz's work, there were two prevailing opinions in the world of science regarding the possibility of accurately forecasting weather over an infinite period of time.
The first approach was formulated back in 1776 by the French mathematician Pierre Simon Laplace, who stated:
...if we imagine a mind that at a given moment comprehends all the connections between objects in the Universe, then it will be able to establish the corresponding position, movements and general effects of all these objects at any time in the past or in the future.
This approach of his was very similar to the famous words of Archimedes: “Give me a fulcrum, and I will turn the whole world upside down.” Thus, Laplace and his supporters argued that for accurate prediction it is necessary to collect as much information as possible about all the particles in the Universe, their location, speed, mass, direction of movement, acceleration, etc. Laplace thought that the more a person knows, the more accurate his forecast will be.
The second approach to the possibility of weather forecasting was most clearly formulated before anyone else by another French mathematician - Jules Henri Poincare (Poincaré). In 1903 he stated:
If we knew exactly the laws of nature and the position of the Universe at the initial moment, we could accurately predict the position of the same Universe at a subsequent moment. But even if the laws of nature revealed all their secrets to us, we would still be able to know the initial position only approximately. If this enabled us to predict the subsequent situation to the same approximation, that would be all we required, and we could say that the phenomenon had been predicted, that it was governed by laws. But this is not always the case: small differences in initial conditions can cause very large differences in the final phenomenon. A small mistake in the former will give rise to a huge mistake in the latter. Prediction becomes impossible, and we are dealing with a phenomenon that develops by chance.
In these words of Poincaré we find the postulate of chaos theory about dependence on initial conditions. The subsequent development of science, especially quantum mechanics, refuted Laplace's determinism.
In 1927, German physicist Werner Heisenberg discovered and formulated the uncertainty principle. This principle explains why some random phenomena do not obey Laplacian determinism. Heisenberg demonstrated the uncertainty principle using the example of radioactive nuclear decay. Thus, due to the very small size of the nucleus, it is impossible to know all the processes occurring inside it. Therefore, no matter how much information we collect about the nucleus, it is impossible to accurately predict when this nucleus will decay.
What tools does chaos theory have? First of all, these are attractors and fractals.
An attractor (from the English to attract - to attract) is a geometric structure that characterizes behavior in phase space after a long time. We can also say that an attractor is the limit of the system, the limit of its oscillations and dynamics.
Here the need arises to define the concept of phase space. So, phase space is an abstract space whose coordinates are the degrees of freedom of the system. For example, the motion of a pendulum has two degrees of freedom. This movement is completely determined by the initial speed of the pendulum and its position. If there is no resistance to the movement of the pendulum, then the phase space will be a closed circle. In reality on Earth, the movement of a pendulum is influenced by the force of friction. In this case, the phase space will be a spiral (Fig. 6.5).
.
In other words, an attractor is the area of solutions, what the system strives to achieve, what it is attracted to.
The simplest type of attractor is a point. Such an attractor is characteristic of a pendulum in the presence of friction. Regardless of the initial speed and position, the pendulum will always come to rest, i.e. exactly.
The next type of attractor can be called a limit cycle, which has the form of a closed curved line. An example of such an attractor would be a pendulum, which is not affected by friction.
Another example of a limit cycle is the beating of the heart. The beat frequency can decrease and increase, but it always tends to its attractor, its closed curve.
The third type of attractor is a torus. In Fig. 6.6 the torus is shown in the upper right corner.
Despite the complexity of the behavior of chaotic attractors, sometimes called strange attractors, knowledge of the phase space makes it possible to represent the behavior of the system in geometric form and predict it accordingly. And although it is almost impossible for the system to be located at a specific moment in time at a specific point in phase space, the area where the object is located and its tendency towards the attractor are predictable.
The first chaotic attractor was the Lorentz attractor (in the lower left corner in Fig. 6.6 and in all its glory in Fig. 6.7).
The Lorentz attractor is calculated on the basis of only three degrees of freedom - three ordinary differential equations, three constants and three initial conditions. However, despite its simplicity, the Lorentz system behaves in a pseudo-random (chaotic) manner.
Br />By simulating his system on a computer, Lorenz identified the reason for its chaotic behavior - the difference in the initial conditions. Even a microscopic deviation of two systems at the very beginning in the process of evolution led to an exponential accumulation of errors and, accordingly, their stochastic divergence.
At the same time, any attractor has limiting dimensions, so the exponential divergence of two trajectories of different systems cannot continue indefinitely. Sooner or later, the orbits will converge again and pass next to each other or even coincide, although the latter is unlikely. By the way, the coincidence of trajectories is a rule of behavior of simple predictable attractors.
Convergence-divergence (also called folding and stretching, respectively) of a chaotic attractor systematically eliminates the initial information and replaces it with new information. As the trajectories converge, the myopia effect begins to appear—the uncertainty of large-scale information increases. When trajectories diverge, the opposite is true - they diverge and the effect of farsightedness appears when the uncertainty of small-scale information increases.
As a result of the constant convergence and divergence of a chaotic attractor, uncertainty rapidly increases, which deprives us of the opportunity to make accurate predictions. What science is so proud of - the ability to establish connections between causes and effects - is impossible in chaotic systems. There is no cause-and-effect relationship between the past and the future in chaos.
It should be noted that the speed of convergence-divergence is a measure of chaos, i.e. a numerical expression of how chaotic the system is. Another statistical measure of chaos is the dimension of the attractor.
Thus, it can be noted that the main property of chaotic attractors is the convergence and divergence of the trajectories of different systems, which are randomly mixed gradually and infinitely.
The intersection of fractal geometry and chaos theory is evident here. And, although one of the tools of chaos theory is fractal geometry, which allows one to obtain complex figures by applying simple rules, a fractal is the opposite of chaos.
The main difference between chaos and fractal is that the former is a dynamic phenomenon, while a fractal is static. The dynamic property of chaos is understood as an unstable and non-periodic change in trajectories.
Introduction to Chaos Theory
What is chaos theory?
Chaos theory is the study of complex nonlinear dynamic systems.
Formally, chaos theory is defined as the study of complex nonlinear dynamic systems. This is what is meant by the term complex, and by the term nonlinear we mean recursion and algorithms from higher mathematics, and, finally, dynamic means non-constant and non-periodic. Thus, chaos theory is the study of constantly changing complex systems, based on non-mathematical concepts of recursion, whether in the form of a recursive process or a set of differential equations that model a physical system.
Misconceptions about Chaos Theory
The general public began to pay attention to chaos theory thanks to films such as Jurassic Park, and thanks to them, public fear of chaos theory is constantly increasing. However, as with anything covered in the media, there are many misconceptions surrounding chaos theory.
The most common discrepancy is that people think that chaos theory is a theory about disorder. Nothing could be further from the truth! This is not a refutation of determinism or a claim that ordered systems are impossible; this is not a denial of experimental evidence or a statement that complex systems are useless. Chaos in chaos theory is order - and not even just order, but the essence of order.
It is true that chaos theory states that small changes can produce huge consequences. But one of the central concepts in the theory is the impossibility of accurately predicting the state of a system. In general, the task of modeling the overall behavior of a system is quite feasible, even simple. Thus, chaos theory focuses its efforts not on the disorder of the system - the hereditary unpredictability of the system - but on the order it inherited - the common behavior of similar systems.
Thus, it would be incorrect to say that chaos theory is about disorder. To illustrate this with an example, let's take the Lorentz attractor. It is based on three differential equations, three constants and three initial conditions.
Chaos theory about disorder
An attractor represents the behavior of a gas at any given time, and its state at a given moment depends on its state at times preceding that moment. If the original data is changed by even very small amounts, say these values are small enough to be comparable to the contribution of individual atoms to Avogadro's number (which is a very small number compared to values on the order of 1024), checking the state of the attractor will show completely different numbers. This happens because small differences are magnified by recursion.
However, despite this, the attractor graph will look quite similar. Both systems will have completely different values at any given time, but the attractor graph will remain the same because it expresses the general behavior of the system.
Chaos theory says that complex nonlinear systems are inherently unpredictable, but at the same time, chaos theory says that the way to express such unpredictable systems turns out to be correct not in exact equalities, but in representations of the system's behavior - in strange attractor graphs or in fractals. Thus, chaos theory, which many people think of as unpredictability, turns out to be, at the same time, the science of predictability even in the most unstable systems.
Application of chaos theory in the real world
When new theories appear, everyone wants to know what's good about them. So what's good about chaos theory? First and most important, chaos theory is a theory. This means that most of it is used more as a scientific basis than as directly applicable knowledge. Chaos theory is a very good way to look at events happening in the world differently from the more traditional clearly deterministic view that has dominated science since Newton. Viewers who have seen Jurassic Park no doubt fear that chaos theory can greatly influence human perception of the world, and, in fact, chaos theory is useful as a means of interpreting scientific data in new ways. Instead of traditional X-Y plots, scientists can now interpret phase-space diagrams that - rather than describing the exact position of any variable at a particular point in time - represent the overall behavior of a system. Instead of looking at exact equalities based on statistical data, we can now look at dynamic systems with behavior similar in nature to static data - i.e. systems with similar attractors. Chaos theory provides a strong framework for the development of scientific knowledge.
However, according to the above, it does not follow that chaos theory has no applications in real life.
Chaos theory techniques have been used to model biological systems, which are undoubtedly some of the most chaotic systems imaginable. Dynamic equation systems have been used to model everything from population growth and epidemics to arrhythmic heartbeats.
In reality, almost any chaotic system can be modeled - the stock market produces curves that can be easily analyzed using strange attractors as opposed to exact relationships; the process of droplets falling from a leaky faucet appears random when analyzed by the naked ear, but when depicted as a strange attractor, it reveals an uncanny order that would not be expected from traditional means.
Fractals are everywhere, most prominently in graphics programs such as the highly successful Fractal Design Painter series of products. Fractal data compression techniques are still being developed, but promise amazing results such as compression ratios of 600:1. The film special effects industry would have much less realistic landscape elements (clouds, rocks and shadows) without fractal graphics technology.
In physics, fractals naturally arise when modeling nonlinear processes, such as turbulent fluid flow, complex diffusion-adsorption processes, flames, clouds, etc. Fractals are used when modeling porous materials, for example, in petrochemistry. In biology, they are used to model populations and to describe internal organ systems (the blood vessel system).
And, of course, chaos theory gives people a surprisingly interesting way to gain an interest in mathematics, one of the least popular areas of knowledge today.
Chaos theory has recently become one of the most fashionable approaches to market research. Unfortunately, an exact mathematical definition of the concept of chaos does not yet exist. Now chaos is often defined as the extreme unpredictability of constant nonlinear and irregular complex motion that occurs in a dynamic system.
CHAOS IS NOT RANDOM
It should be noted that chaos is not random, despite its unpredictability. Moreover, chaos is dynamically determined (determined). At first glance, unpredictability borders on randomness - after all, we, as a rule, cannot predict just random phenomena.
And if you treat the market as a random walk, then this is exactly the case. However, chaos is not accidental; it obeys its own laws. According to chaos theory, if you talk about chaotic price movement, then you should not mean a random price movement, but another, especially ordered movement. If market dynamics are chaotic, they are not random, although they are still unpredictable.
The unpredictability of chaos
The unpredictability of chaos is explained mainly by its significant dependence on the initial conditions. This dependence indicates that even the smallest errors in measuring the parameters of the object under study can lead to completely incorrect predictions.
These errors can arise due to elementary ignorance of all initial conditions. Something will definitely escape our attention, which means that already in the very formulation of the problem there will be an internal error, which will lead to significant errors in predictions.
"Butterfly Effect"
In relation to the inability to make long-term weather forecasts, the significant dependence on initial conditions is sometimes called the “butterfly effect.” The "butterfly effect" refers to the possibility that the flapping of a butterfly's wing in Brazil will result in a tornado in Texas.
Additional inaccuracies in the results of research and calculations can be introduced by the most invisible factors influencing the system at first glance, which appear during the period of its existence from the initial moment until the appearance of the actual and final result. In this case, the influencing factors can be both exogenous (external) and endogenous (internal).
A striking example of chaotic behavior is the movement of a billiard ball. If you have ever played billiards, then you know that the final result depends on the initial accuracy of the hit, its force, the position of the cue relative to the ball, the assessment of the location of the ball being hit, as well as the location of other balls on the table. The slightest inaccuracy in one of these factors leads to the most unpredictable consequences - the ball may roll completely differently from where the billiard player expected. Moreover, even if the billiard player did everything correctly, try to predict the movements of the ball after five or six collisions.
Let's consider another example of the influence of initial conditions on the final result. Let's imagine, for example, a stone on top of a mountain. Just push him a little, and he will roll down to the very foot of the mountain. It is clear that a very small change in the force of the push and its direction can lead to a very significant change in where the stone stops at the foot. There is, however, one very significant difference between the example with a stone and a chaotic system.
In the first, the factors influencing the stone during its fall from the mountain (wind, obstacles, changes in the internal structure due to collisions, etc.) no longer have a strong impact on the final result compared to the initial conditions. In chaotic systems, small changes have a significant impact on the outcome not only in the initial conditions, but also in other factors.
One of the main conclusions of chaos theory, therefore, is the following - it is impossible to predict the future, since there will always be measurement errors, generated, among other things, by ignorance of all factors and conditions.
The same thing is simple - small changes and/or mistakes can have big consequences.
Figure 1. Significant dependence of the result on the initial conditions and influencing factors
- Another basic property of chaos is the exponential accumulation of error. According to quantum mechanics, the initial conditions are always uncertain, and according to chaos theory, these uncertainties will quickly grow and exceed the permissible limits of predictability.
- The second conclusion of chaos theory is that the reliability of forecasts quickly decreases over time.
Figure 2. Exponential decline in forecast confidence
It is usually said that chaos is a higher form of order, but it is more correct to consider chaos as another form of order - inevitably in any dynamic system, order in the usual sense is followed by chaos, and chaos is followed by order. If we define chaos as disorder, then in such disorder we will definitely be able to see our own special form of order. For example, smoke from cigarettes, initially rising in the form of an ordered column, under the influence of the external environment, takes on more and more bizarre shapes, and its movements become chaotic.Another example of randomness in nature is a leaf from any tree. It can be argued that you will find many similar leaves, for example oak, but not a single pair of identical leaves. The difference is predetermined by temperature, wind, humidity and many other external factors, in addition to purely internal causes (for example, genetic differences).
Movement from order to chaos and back again seems to be the essence of the universe, no matter what manifestations of it we study. Even in the human brain there is both order and chaos at the same time. The first corresponds to the left hemisphere of the brain, and the second to the right. The left hemisphere is responsible for conscious human behavior, for the development of linear rules and strategies in human behavior, where “if... then...” is clearly defined. In the right hemisphere, nonlinearity and chaos reign. Intuition is one of the manifestations of the right hemisphere of the brain.
Chaos theory studies the order of a chaotic system, which appears random, disordered. At the same time, chaos theory helps to build a model of such a system, without setting the task of accurately predicting the behavior of a chaotic system in the future.
The first elements of chaos theory appeared in the 19th century, but this theory received its true scientific development in the second half of the 20th century, together with the work of Edward Lorenz of the Massachusetts Institute of Technology and the French-American mathematician Benoit B. Mandelbrot ).
Edward Lorenz at one time (early 60s of the 20th century, work published in 1963) looked at the difficulties in weather forecasting.
Before Lorenz's work, there were two prevailing opinions in the world of science regarding the possibility of accurately forecasting weather over an infinite period of time.
The first approach was formulated back in 1776 by the French mathematician Pierre Simon Laplace. Laplace stated that"...if we imagine a mind that at a given moment has comprehended all the connections between objects in the Universe, then it will be able to establish the corresponding position, movements and general effects of all these objects at any time in the past or in the future". This approach of his was very similar to the famous words of Archimedes: “Give me a fulcrum, and I will turn the whole world upside down.”
Thus, Laplace and his supporters said that to accurately predict the weather, it is only necessary to collect more information about all the particles in the Universe, their location, speed, mass, direction of movement, acceleration, etc. Laplace thought that the more a person knows, the more accurate his forecast about the future will be.
The second approach to the possibility of weather forecasting was most clearly formulated before anyone else by another French mathematician, Jules Henri Poincaré. In 1903 he said:“If we knew exactly the laws of nature and the position of the Universe at the initial moment, we could accurately predict the position of the same Universe at a subsequent moment. But even if the laws of nature revealed to us all their secrets, we could then only know the initial position approximately . If this enabled us to predict the subsequent situation with the same approximation, that would be all that we require, and we could say that the phenomenon was predicted, that it was governed by laws. But this is not always the case; it may happen that small differences in the initial conditions will produce very large differences in the final phenomenon. A small error in the former will produce a huge error in the latter. Prediction becomes impossible, and we are dealing with a phenomenon that develops by chance."
In these words of Poincaré we find the postulate of chaos theory about dependence on initial conditions. The subsequent development of science, especially quantum mechanics, refuted Laplace's determinism. In 1927, German physicist Werner Heisenberg discovered and formulated the uncertainty principle. This principle explains why some random phenomena do not obey Laplacian determinism. Heisenberg demonstrated the uncertainty principle using the example of radioactive nuclear decay. Thus, due to the very small size of the nucleus, it is impossible to know all the processes occurring inside it. Therefore, no matter how much information we collect about the nucleus, it is impossible to accurately predict when this nucleus will decay.
What tools does chaos theory have? First of all, these are attractors and fractals.
Attractor(from English to attract- attract) is a geometric structure that characterizes behavior in phase space after a long time.
Here it becomes necessary to define the concept of phase space. So, phase space is an abstract space whose coordinates are the degrees of freedom of the system. For example, the motion of a pendulum has two degrees of freedom. This movement is completely determined by the pendulum's initial speed and position.
If there is no resistance to the movement of the pendulum, then the phase space will be a closed curve. In reality on Earth, the movement of a pendulum is influenced by the force of friction. In this case, the phase space will be a spiral.
Figure 3. Pendulum motion as an example of phase space
Simply put, an attractor is what the system strives to achieve, what it is attracted to.
- The simplest type of attractor is a point. Such an attractor is characteristic of a pendulum in the presence of friction. Regardless of the initial speed and position, such a pendulum will always come to rest, i.e. exactly.
- The next type of attractor is a limit cycle, which has the form of a closed curved line. An example of such an attractor is a pendulum, which is not affected by friction. Another example of a limit cycle is the beating of the heart. The beat frequency can decrease and increase, but it always tends to its attractor, its closed curve.
- The third type of attractor is a torus. In Figure 4, the torus is shown in the upper right corner.
Figure 4. Main types of attractors. Shown above are three predictable, simple attractors. Below are three chaotic attractors.
Despite the complexity of the behavior of chaotic attractors, sometimes called strange attractors, knowledge of the phase space makes it possible to represent the behavior of the system in geometric form and predict it accordingly.And although it is almost impossible for the system to be located at a specific moment in time at a specific point in phase space, the area where the object is located and its tendency towards the attractor are predictable.
The first chaotic attractor was the Lorentz attractor. In Figure 3.7. it is shown in the lower left corner.
Figure 5. Chaotic Lorenz attractor
The Lorentz attractor is calculated based on only three degrees of freedom - three ordinary differential equations, three constants and three initial conditions. However, despite its simplicity, the Lorentz system behaves in a pseudo-random (chaotic) manner.Having simulated his system on a computer, Lorenz identified the reason for its chaotic behavior - the difference in the initial conditions. Even a microscopic deviation of two systems at the very beginning in the process of evolution led to an exponential accumulation of errors and, accordingly, their stochastic divergence.
At the same time, any attractor has limiting dimensions, so the exponential divergence of two trajectories of different systems cannot continue indefinitely. Sooner or later the orbits will converge again and pass next to each other or even coincide, although the latter is very unlikely. By the way, the coincidence of trajectories is a rule of behavior of simple predictable attractors.
Convergence-divergence (also called folding and stretching, respectively) of a chaotic attractor systematically eliminates the initial information and replaces it with new information. As the trajectories converge, the myopia effect begins to appear—the uncertainty of large-scale information increases. When trajectories diverge, on the contrary, they diverge and the effect of farsightedness appears when the uncertainty of small-scale information increases.
As a result of the constant convergence and divergence of a chaotic attractor, uncertainty rapidly increases, which with each moment in time deprives us of the opportunity to make accurate forecasts. What science is so proud of - the ability to establish connections between causes and effects - is impossible in chaotic systems. There is no cause-and-effect relationship between the past and the future in chaos.
It should be noted here that the speed of convergence-divergence is a measure of chaos, i.e. a numerical expression of how chaotic the system is. Another statistical measure of chaos is the dimension of the attractor.
Thus, it can be noted that the main property of chaotic attractors is the convergence-divergence of trajectories of different systems, which are randomly mixed gradually and infinitely
The intersection of fractal geometry and chaos theory is evident here. And, although one of the tools of chaos theory isfractal geometry, fractal is the opposite of chaos.
The main difference between chaos and fractal is that the former is a dynamic phenomenon, while a fractal is static. The dynamic property of chaos is understood as an unstable and non-periodic change in trajectories.
FRACTAL
Fractal- this is a geometric figure, a certain part of which is repeated again and again, hence one of the properties of a fractal is manifested - self-similarity.Another property of a fractal is fractionality. The fractionality of a fractal is a mathematical reflection of the degree of irregularity of the fractal.
In fact, anything that seems random and irregular can be a fractal, such as clouds, trees, river bends, heartbeats, animal populations and migrations, or flames.
Figure 6. Sierpinski carpet fractal
This fractal is obtained through a series of iterations. Iteration (from Latin iteratio - repetition) is the repeated application of any mathematical operation.
Figure 7. Construction of a Sierpinski carpet
A chaotic attractor is a fractal. Why? In a strange attractor, as well as in a fractal, as it increases, more and more details are revealed, i.e. The principle of self-similarity is triggered. No matter how we change the size of the attractor, it will always remain proportionally the same.
In technical analysis, a typical example of a fractal is Elliott waves, where the principle of self-similarity also works.
The first most famous and authoritative scientist to study fractals was Benoit Mandelbrot. In the mid-60s of the 20th century, he developed fractal geometry or, as he also called it, the geometry of nature. Mandelbrot wrote about this in his famous work “Fractal Geometry of Nature”(The Fractal Geometry of Nature). Many people call Mandelbrot the father of fractals, because... he was the first to use it in relation to the analysis of fuzzy, irregular forms.
An additional idea inherent in fractality is non-integer dimensions. We usually talk about one-dimensional, two-dimensional, three-dimensional, etc. integer world. However, there may also be non-integer dimensions, for example, 2.72. Mandelbrot calls such dimensions fractal dimensions.
The logic of the existence of non-integer dimensions is very simple. Thus, in nature there is hardly an ideal ball or cube, therefore, the 3-dimensional dimension of this real ball or cube is impossible and other dimensions must exist to describe such objects.
It is to measure such irregular, fractal figures that the concept of fractal measurement was introduced. For example, crumple a piece of paper into a ball. From the point of view of classical Euclidean geometry, the newly formed object will be a three-dimensional ball. However, in reality it is still just a two-dimensional piece of paper, albeit crumpled into a ball. From this we can assume that the new object will have a dimension greater than 2, but less than 3. This does not fit well with Euclidean geometry, but can be well described using fractal geometry, which would state that the new object will be in a fractal dimension approximately equal to 2.5, i.e. have a fractal dimension of about 2.5.
Deterministic fractals
There are deterministic fractals, an example of which is the Sierpinski carpet, and complex fractals. When constructing the former, no formulas or equations are needed. It is enough to take a sheet of paper and carry out several iterations on some shape. Complex fractals have infinite complexity, although they are generated by a simple formula.A classic example of a complex fractal is the set
Mandelbrot, obtained from the simple formula Zn+1=Zna+C, where Z and C are complex numbers and a is a positive number. In Figure 8 we see a fractal of the 2nd degree, where a = 2.Figure 8. Mandelbrot set
Systems can transition to chaos in different ways. Among the latter, bifurcations are distinguished, which are studied by the theory of bifurcations.
Bifurcation (from lat. bifurcus- bifurcated) is a process of qualitative transition from a state of equilibrium to chaos through a successive very small change (for example, Feigenbaum doubling during a doubling bifurcation) of periodic points.
It is imperative to note what is happeningqualitychanging the properties of the system, the so-called. catastrophic jump. The moment of the jump (bifurcation during doubling bifurcation) occurs atbifurcation point.
Chaos can arise through bifurcation, as shown by Mitchell Feigenbaum. When creating his own theory of fractals, Feigenbaum mainly analyzed the logistic equation Xn+1=CXn - C(Xn)2, where C is an external parameter, from which he concluded that under certain restrictions in all such equations there is a transition from an equilibrium state to chaos .
Below is a classic biological example of this equation.
For example, a population of individuals with a normalized size Xn lives in isolation. A year later, offspring numbering Xn+1 appear. Population growth is described by the first term on the right side of the equation (СХn), where the coefficient C determines the growth rate and is the determining parameter. The loss of animals (due to overpopulation, lack of food, etc.) is determined by the second, nonlinear term (C(Xn)2).
The result of the calculations is the following conclusions:
- at C< 1 популяция с ростом n вымирает;
- in area 1< С < 3 численность популяции приближается к постоянному значению Х0 = 1 - 1/С, что является областью стационарных, фиксированных решений. При значении C = 3 точка бифуркации становится repulsive fixed dot. From this point on, the function never converges to one point. Before this point was attractive fixed;
- in range 3< С < 3.57 начинают появляться бифуркации и разветвление каждой кривой на две. Здесь функция (численность популяции) колеблется между двумя значениями, лежащими на этих ветвях. Сначала популяция резко возрастает, на следующий год возникает перенаселенность и через год численность снова уменьшается;
- at C > 3.57, the areas of different solutions overlap (they seem to be painted over) and the behavior of the system becomes chaotic.
The dependence of population size on parameter C is shown in the following figure.
Figure 9. Transition to chaos through bifurcations, initial stage of the equation Xn+1=CXn - C(Xn)2
Dynamic variables Xn take values that strongly depend on the initial conditions. When calculations are carried out on a computer, even for very close initial values of C, the final values can differ sharply. Moreover, the calculations become incorrect, since they begin to depend on random processes in the computer itself (voltage surges, etc.).
Thus, the state of the system at the moment of bifurcation is extremely unstable and an infinitesimal impact can lead to the choice of a further path of movement, and this, as we already know, is the main feature of a chaotic system (significant dependence on the initial conditions).
Feigenbaum established universal laws of transition to dynamic chaos when the period is doubled, which were experimentally confirmed for a wide class of mechanical, hydrodynamic, chemical and other systems. The result of Feigenbaum's research was the so-called. "Feigenbaum tree".
Figure 10. Feigenbaum tree (calculation based on a slightly modified logistic formula)
What are bifurcations in everyday life, in simple terms. As we know from the definition, bifurcations occur when a system transitions from a state of apparent stability and equilibrium to chaos.
Examples of such transitions are smoke, water and many other common natural phenomena. Thus, the smoke rising upward initially looks like an orderly column. However, after some time, it begins to undergo changes that at first seem orderly, but then become chaotically unpredictable.
In fact, the first transition from stability to some form of apparent orderliness, but already variability, occurs at the first bifurcation point. Further, the number of bifurcations increases, reaching enormous values. With each bifurcation, the smoke turbulence function approaches chaos.
Using the theory of bifurcations, it is possible to predict the nature of the movement that occurs during the transition of a system to a qualitatively different state, as well as the region of existence of the system and evaluate its stability.
Unfortunately, the very existence of chaos theory is difficult to reconcile with classical science. Typically, scientific ideas are tested by making predictions and checking them against actual results. However, as we already know, chaos is unpredictable; when you study a chaotic system, you can only predict its behavior model.
Therefore, with the help of chaos, it is not only impossible to construct an accurate forecast, but also, accordingly, to check it. However, this should not mean that the chaos theory, confirmed both in mathematical calculations and in life, is incorrect.
At present, there is no mathematically precise apparatus for applying chaos theory to study market prices, so there is no rush to apply knowledge about chaos. At the same time, this is truly the most promising modern area of mathematics from the point of view of applied research in financial markets.
The theory of “controlled chaos” is a modern phenomenon, a geopolitical doctrine rooted in ancient sciences such as philosophy, mathematics, and physics. The concept of “chaos” arose from the name in ancient Greek mythology of the original state of the world, a certain “opening abyss” from which the first deities arose.
Attempts to scientifically comprehend the concepts of “order” and “chaos” have formed theories of directed disorder, extensive classifications and typologies of chaos. In the most ancient historical and philosophical tradition, chaos was understood as an all-embracing and generative principle. In the ancient worldview, formless and incomprehensible chaos is endowed with formative power and signifies the primary formless state of matter and the primary potency of the world.
The current level of scientific research has based chaos theory on the assertion that complex systems are extremely dependent on initial conditions, and small changes in the environment can lead to unpredictable consequences.
Stephen Mann is a key figure in the development of the geopolitical doctrine of “chaos management,” including within the framework of US national interests. Stephen Mann (born 1951) graduated from Oberlin College in 1973 (BA in German), received a MA in German literature from Cornwall University (New York) in 1974, and has been in the diplomatic service since 1976. He began his career as an employee of the US Embassy in Jamaica. Then he worked in Moscow and in the Office of Soviet Affairs at the State Department in Washington, worked in the State Department Operations Center (a 24-hour crisis center), and also from 1991 to 1992. - in the office of the Secretary of Defense, covering issues of Russia and Eastern Europe. In 1985-1986 was a fellow at the Harriman Institute for Advanced Soviet Studies at Columbia University (where he received his master's degree in political science). He was the first US Chargé d'Affaires to Micronesia (1986-1988), Mongolia (1988) and Armenia (1992). In 1991, he graduated with honors from the National War College in Washington. In 1992-1994. was Deputy Ambassador to Sri Lanka. In 1995-1998 served as Director of the India, Nepal and Sri Lanka Division at the US State Department. From 1998 to May 2001, he served as US Ambassador to Turkmenistan. Since May 2001, Stephen Mann has been the special representative of the US President for the countries of the Caspian Basin. He is the main representative of American energy interests in this region, a lobbyist for the ABTD project (Aktau-Baku-Tbilisi-Ceyhan oil pipeline).
Based on the results of his studies at the National War College, Stephen Mann in 1992 prepared an article that received great resonance in the military-political community: “Chaos Theory and Strategic Thought.” It was published in the main professional journal of the US Army (Mann, Steven R. Chaos Theory and Strategic Thought // Parameters (US Army War College Quarterly), Vol. XXII, Autumn 1992, pp. 54-68).
In this article, S. Mann makes the following points: “We can learn a lot from seeing chaos and regrouping as opportunities, rather than rushing towards stability as an illusory goal...”. "The international environment is an excellent example of a chaotic system... 'self-organized criticality'... corresponds to it as a means of analysis... The world is destined to be chaotic because the diverse actors of human politics in a dynamic system... have different goals and values." . “Every actor in politically critical systems produces the energy of conflict ... which provokes a change in the status quo, thus participating in the creation of a critical state ... and any course leads the state of affairs to an inevitable cataclysmic reorganization.”
The main idea that follows from Mann’s presented theses is to transfer the system to a state of “political criticality.” And then, under certain conditions, it will inevitably plunge itself into cataclysms of chaos and “reorganization.” In the context of his article, it is important to note that the approach in question can be used both for social creation and for asocial destruction and geopolitical manipulation.
It is absolutely clear from S. Mann's report that not only scientific and ideological thought is evident, but also the pursuit of US national security. In the article, Mann writes: “With American advantages in communications and increasing opportunities for global travel, the virus (we are talking about “ideological contagion”) will be self-perpetuating and will spread in a chaotic manner. Therefore, our national security will have the best guarantees..." And further: “This is the only way to build a long-term world order. If we fail to achieve such ideological change throughout the world, we will be left with sporadic periods of calm between catastrophic realignments.” Mann's words about “world order” here are a tribute to “political correctness.” Because his report speaks exclusively of chaos, in which, judging by Mann’s words about “the best guarantees of US national security,” only America will have the opportunity to survive as an “island of order” in an ocean of “controlled criticality” or global chaos.