Quite wide and free. Such a formal transfer of meaning from the natural sciences to the humanities often leads to the substitution of concepts. Meanwhile, this rather specific term has a special meaning, which, however, can be interpreted depending on the context.
The word "bifurcation" comes from the Latin term for bifurcation. It is used in natural sciences when they want to describe the qualitative restructuring of an object and the metamorphoses associated with it.
When a system evolves, its state depends on one or more parameters that can change smoothly. But sometimes one of the characteristics acquires critical significance, and the system enters the stage of fundamental qualitative change.
The very moment at which the mode of change in the system is rebuilt is called the bifurcation point. And by bifurcation we mean the restructuring of the system itself.
What happens if the system changes continuously? In this case, so-called cascades of bifurcations are observed, which successively replace each other.
The description of these systemic changes represents one of the scenarios of transition from simple to complex, from ordered movement to chaotic.
Bifurcation point as a moment of truth
By describing a system as a sequence of bifurcations replacing one another, it is possible to create a model of the development of any more or less complex system, no matter what field of knowledge it belongs to.
Bifurcation points can be observed not only in biological and physical systems, but also in economic and social systems.
From the point of view of everyday life, the transition of a system through a bifurcation point can be compared with the behavior of a person or a living organism in a situation where only one of many choices is possible. A striking example here is the knight at the crossroads, who stopped thoughtfully in front of a stone with inscriptions.
Two or even three paths open before the thoughtful warrior, each of which has equal significance for the traveler. Which road the knight chooses depends on some
What does he study? bifurcation theory.
Bifurcation
Bifurcation(from the Latin 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 that there is a qualitative change in the properties of the system, the so-called. catastrophic jump. The moment of the jump (splitting at the doubling bifurcation) occurs at the bifurcation point.
Chaos can arise through bifurcation, as shown by Mitchell Feigenbaum. When creating his own, Feigenbaum mainly analyzed the logistic equation:
Xn+1=CXn - C(Xn) 2,
Where WITH— external parameter.
Where does the conclusion come from that, under certain restrictions, in all such equations there is a transition from an equilibrium state to chaos.
Bifurcation example
Below is a classic biological example of this equation.
For example, a population of individuals of normalized size lives in isolation Xn. A year later, offspring numbering Xn +1. Population growth is described by the first term on the right side of the equation (СХn), where coefficient C determines the growth rate and is the determining parameter. Animal damage (due to overcrowding, lack of food, etc.) is determined by the second, nonlinear term C(Xn) 2.
The result of the calculations is the following conclusions:
- At WITH<1 the population dies out as n increases;
- In area 1<С<3 population size is approaching a constant value X0=1-1/C, which is the region of stationary, fixed solutions. When value C=3 the bifurcation point becomes a repulsive fixed point. From this point on, the function never converges to one point. Before this, the point was a fixed attractor;
- In range 3<С
- When C> 3.57, the areas of different solutions overlap (they seem to be painted over) and the behavior of the system becomes chaotic.
Hence the conclusion - the final state of physical systems that evolve is the state dynamic chaos.
Dependence of population size on parameter WITH shown in the following figure.
Figure 1 — 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. "".
Figure 2 - Feigenbaum tree (calculation based on a modified logical formula)
Let us denote by 
the value of the parameter at which period doublings occurred. In 1971, the American scientist M. Feigenbaum established an interesting pattern: the sequence forms an increasing sequence, quickly converges with an accumulation point of 3.5699... The difference in values corresponding to two successive bifurcations decreases each time by approximately the same factor:
The denominator of the progression =4.6692 is now called Feigenbaum constant.
The concept of bifurcation
What are bifurcations in everyday life? As we know from the definition, bifurcations arise during the transition of a system from a state of apparent stability and equilibrium to chaos. Examples of such transitions are smoke, water and many other common natural phenomena. So that the smoke rising upward first looks like an orderly column.
Smoke as an example of the occurrence of bifurcation during the transition of a system from a state of apparent stability and equilibrium to chaos However, after a while it begins to undergo changes, at first appearing orderly, but then becoming 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.
By using bifurcation theory 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. Of course, scientific ideas are tested based on predictions and their comparison with 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 the moment, 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 “strangeness” of a chaotic attractor lies not so much in its unusual appearance, but in the new properties that it possesses. A strange attractor is primarily an attractive region for trajectories from surrounding regions. Moreover, all trajectories inside the strange attractor are dynamically unstable.
In other words, if we imagine the limit set as a “tangle” in phase space, then the point characterizing the state of the system belongs to this “tangle” and will not go to another region of phase space. However, we cannot say where in the ball the point is at a given time.
Positive Lyapunov exponent
One of these paradoxical properties is sensitivity to initial data. Let's illustrate this. Let's choose two close points x"(0) and x"(0), belonging to the attractor trajectory, and see how the distance d(t) = |x"(t) - x"(t) | with time. If the attractor is a singular point, then d(t) = 0. If the attractor is a limit cycle, then d(t) will be a periodic function of time. The lambda value is called Lyapunov exponent. The positive Lyapunov exponent characterizes the average acceleration rate of infinitely close trajectories.
The positive values of the Lyapunov exponent and the sensitivity of the system to the initial data allowed us to take a completely different look at the problem of forecasting. Previously, it was assumed that a forecast of the behavior of deterministic systems, in contrast to stochastic ones, can be given for any desired time.
However, research in recent decades has shown that there is a class of deterministic systems (even relatively simple ones), the behavior of which can be predicted only for a limited period of time. In a strange attractor, after time two initially close trajectories cease to be close. No matter how small the inaccuracy in determining the initial state increases over time, and in principle we cannot give a “long-term forecast.” Thus, there is a forecast horizon that limits our ability to foresee.
Fractal structure
Another interesting characteristic of the chaotic regime is fractal structure. The geometric structure of a strange attractor cannot be represented in the form of curves or planes, or geometric elements of an entire dimension. The dimension of a strange attractor is fractional, or, as they say, fractal.
Bifurcation point- change of the established operating mode of the system. Term from nonequilibrium thermodynamics and synergetics.
Bifurcation point- a critical state of the system, in which the system becomes unstable with respect to fluctuations and uncertainty arises: whether the state of the system will become chaotic or whether it will move to a new, more differentiated and higher level of order. A term from the theory of self-organization.
Properties of a bifurcation point
- Unpredictability. Usually the bifurcation point has several attractor branches (stable operating modes), one of which the system will follow. However, it is impossible to predict in advance which new attractor the system will occupy.
- The bifurcation point is short-term in nature and separates longer stable regimes of the system.
- The avalanche effect of hash functions involves planned bifurcation points that deliberately introduce unpredictable changes to the final form of the hash string when even a single character in the original string changes.
see also
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Literature
- // Lebedev S. A. Philosophy of Science: Dictionary of Basic Terms. - M.: Academic Project, 2004. - 320 p. - (Gaudeamus Series).
Excerpt characterizing the Bifurcation Point
“Cette pauvre armee,” he said suddenly, “elle a bien diminue depuis Smolensk.” La fortune est une franche courtisane, Rapp; je le disais toujours, et je commence a l "eprouver. Mais la garde, Rapp, la garde est intacte? [Poor army! It has greatly diminished since Smolensk. Fortune is a real harlot, Rapp. I have always said this and am beginning to experience it. But the guard, Rapp, are the guards intact?] – he said questioningly.“Oui, Sire, [Yes, sir.],” answered Rapp.
Napoleon took the lozenge, put it in his mouth and looked at his watch. He didn’t want to sleep; morning was still far away; and in order to kill time, no orders could be made anymore, because everything had been done and was now being carried out.
– A t on distribue les biscuits et le riz aux regiments de la garde? [Did they distribute crackers and rice to the guards?] - Napoleon asked sternly.
– Oui, Sire. [Yes, sir.]
– Mais le riz? [But rice?]
Rapp replied that he had conveyed the sovereign’s orders about rice, but Napoleon shook his head with displeasure, as if he did not believe that his order would be carried out. The servant came in with punch. Napoleon ordered another glass to be brought to Rapp and silently took sips from his own.
“I have neither taste nor smell,” he said, sniffing the glass. “I’m tired of this runny nose.” They talk about medicine. What kind of medicine is there when they cannot cure a runny nose? Corvisar gave me these lozenges, but they don't help. What can they treat? It cannot be treated. Notre corps est une machine a vivre. Il est organise pour cela, c"est sa nature; laissez y la vie a son aise, qu"elle s"y defende elle meme: elle fera plus que si vous la paralysiez en l"encombrant de remedes. Notre corps est comme une montre parfaite qui doit aller un certain temps; l"horloger n"a pas la faculte de l"ouvrir, il ne peut la manier qu"a tatons et les yeux bandes. Notre corps est une machine a vivre, voila tout. [Our body is a machine for life. This is what it is designed for. Leave the life in him alone, let her defend herself, she will do more on her own than when you interfere with her with medications. Our body is like a clock that must run for a certain time; the watchmaker cannot open them and can only operate them by touch and blindfolded. Our body is a machine for life. That's all.] - And as if having embarked on the path of definitions, definitions that Napoleon loved, he suddenly made a new definition. – Do you know, Rapp, what the art of war is? - he asked. – The art of being stronger than the enemy at a certain moment. Voila tout. [That's all.]
Dissipative open systems. Bifurcation point.
Open systems in which an increase in entropy is observed are called dissipative. In such systems, the energy of ordered motion transforms into the energy of disordered chaotic motion, into heat. If a closed system (Hamiltonian system), removed from a state of equilibrium, always tends to return to a maximum of entropy, then in an open system the outflow of entropy can balance its growth in the system itself and there is a possibility of a stationary state occurring. If the outflow of entropy exceeds its internal growth, then large-scale fluctuations arise and grow to the macroscopic level, and under certain conditions, self-organizing processes and the creation of ordered structures begin to occur in the system.
When studying systems, they are often described by a system of differential equations. The representation of the solution to these equations as the movement of a certain point in space with a dimension equal to the number of variables is called phase trajectories of the system. The behavior of the phase trajectory in terms of stability shows that there are several main types of it, when all solutions of the system ultimately focus on a certain subset. Such a subset is called attractor. Attractor has a region of attraction, a set of initial points, such that as time increases, all phase trajectories that begin in them tend precisely to this attractor.
The main types of attractors are:
stable limit points
· stable cycles (the trajectory tends to some closed curve)
· tori (to the surface of which the trajectory approaches)
The motion of a point in such cases has a periodic or quasiperiodic character. There are also so-called strange attractors characteristic only of dissipative systems, which, unlike ordinary ones, are not submanifolds of phase space (a point, a cycle, a torus, a hypertorus are) and the movement of a point on them is unstable, any two trajectories on it always diverge, a small change in the initial data leads to different development paths. In other words, the dynamics of systems with strange attractors are chaotic.
Equations with strange attractors are not exotic at all. An example of such a system is the Lorentz system, obtained from the hydrodynamic equations in the problem of thermoconvection of a liquid layer heated from below.
The structure of strange attractors is remarkable. Their unique property is the scaling structure or large-scale self-repeatability. This means that by enlarging a section of an attractor containing an infinite number of curves, one can verify its similarity to a large-scale representation of part of the attractor. For objects that have the ability to endlessly repeat their own structure at the micro level, there is a special name - fractals.
Dynamic systems that depend on a certain parameter are, as a rule, characterized by a smooth change in the nature of behavior when the parameter changes. However, the parameter may have some critical (bifurcation) value, upon passing through which the attractor undergoes a qualitative restructuring and, accordingly, the dynamics of the system changes sharply, for example, stability is lost. Loss of stability occurs, as a rule, through a transition from a point of stability to a stable cycle (soft loss of stability), exit of the trajectory from a stable position (hard loss of stability), and the birth of cycles with a double period. With further changes in the parameter, tori and then strange attractors, that is, chaotic processes, may appear.
Here it must be stated that in the special sense of the word chaos means irregular motion described by deterministic equations. Irregular movement implies the impossibility of describing it by the sum of harmonic movements.
Bifurcation point- one of the most significant concepts in the theory of self-organization. This is a period or moment in the history of a system when it transforms from one systemic certainty to another. Its qualitative characteristics after reaching the bifurcation point are doomed to a fundamental change, leading to a change in the essence of the system itself. The system transformation mechanism that operates at such moments is associated with the branching of the system trajectory, determined by the presence of competition among attractors.
Bifurcation points- special moments in the development of living and non-living systems, when sustainable development, the ability to suppress random deviations from the main direction, are replaced by instability. Two or more (instead of one) new states become stable. The choice between them is determined by chance, in the phenomena of social life - by a volitional decision. After making a choice, self-regulatory mechanisms maintain the system in one state (on one trajectory), the transition to another trajectory becomes difficult. For example, the evolution of living organisms and the emergence of new species fit completely into this scheme. As conditions change, a previously well-adapted species loses stability, and as a result of bifurcation, two new species differ from the previous one, and to an even greater extent - from each other. Examples of bifurcation points: freezing of supercooled water; changing the political structure of the state through revolution.
Bifurcation point- a period in the development of the system when the previous stable, linear and predictable path of development of the system becomes impossible, this is a point of critical instability of development, in which the system is rebuilt, chooses one of the possible paths of further development, that is, a certain phase transition occurs.
Examples of bifurcation in various systems the following can serve: river bifurcation - division of a river bed and its valley into two branches, which subsequently do not merge and flow into different basins; in medicine - the division of a tubular organ (vessel or bronchus) into 2 branches of the same caliber, extending to the sides at the same angles; mechanical bifurcation - the acquisition of a new quality in the movements of a dynamic system with a small change in its parameters; in the education system - division of the senior classes of an educational institution into two departments; time-space bifurcation (in science fiction) - division of time into several streams, each of which has its own events. In parallel time-space, the heroes have different lives.