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The point of bifurcation is a change in the steady-state mode of operation of the system

Modern popular science and popular literature often uses the terms "synergetics", "chaos theory" and "bifurcation point". This new trend of populist use of the theory of complex systems often replaces the conceptual and contextual meaning of definitions. Let's try not abstrusely, but nevertheless close to the scientific one to explain to the reader interested in the meaning and essence of these concepts.

Science and self-organizing systems

Interdisciplinary teaching, exploring regularities in complex systems of any nature is synergetics. The point of bifurcation as a turning point or the moment of choice is a key concept in the theory of the behavior of complex systems. The synergetic concept of complex systems implies their openness (the exchange of matter, energy, information with the environment), the nonlinearity of development (the existence of multiple paths of development), dissipativity (the discharge of excess entropy), and the possibility of a state of bifurcation (choice or crisis point). The synergetic theory is applicable to all systems where there is a sequence and spasmodic changes developing in time - biological, social, economic, physical.

The donkey of Buridan

A common technique is to explain the complicated by simple examples. The classics of the illustration describing the state of a system approaching a bifurcation point is an example of a well-known 14th century logician Jean Buridan with an ass, his master and philosopher. The initial tasks are as follows. There is a choice - two piles of hay. There is an open system - an ass, located at the same distance from both haystacks. Observers are the master of the donkey and philosopher. The question is - what kind of bag of hay will the donkey choose? In Buridan, in a parable for three days, people watched the donkey, who could not make a choice until the owner connected the heaps. And no one died of hunger.

The concept of bifurcation interprets the situation like this. The end of the parable is omitted, we will concentrate on the situation of choice between equilibrium objects. At this point, any change can lead to a shift in the direction of one of the objects (for example, the donkey fell asleep, waking up, was closer to one of the heaps of hay). In synergy, the donkey is a complex open system. The point of bifurcation is the state of the donkey before the equilibrium choice. The change in position is a perturbation (fluctuation) of the system. And two haystacks are attractors, then the state in which the system will come after passing through the point of bifurcation and reaching a new equilibrium state.

Three fundamental points of bifurcation

The state of the system approaching the bifurcation point is characterized by three fundamental components: fracture, choice and ordering. Before the bifurcation point, the system resides in an attractor (a property characterizing the stability of the system). At the bifurcation point, the system is characterized by fluctuations (perturbations, fluctuations in indices) that cause a qualitative and quantitative jump in the system with the choice of a new attractor or transition to a new stable state. The multiplicity of possible attractors and the huge role of chance open up the multivariate organization of the system.

Mathematics describes the bifurcation points and the stages of its passage through the system in complex differential equations with a set of all parameters and fluctuations.

Unpredictable bifurcation point

This is the state of the system before choosing, at a crossroads, at the point of divergence of multiple choice and development options. In the intervals between bifurcations, the linear behavior of the system is predictable, it is determined by both random and regular factors. But at the point of bifurcation, the role of randomness comes out on top, and a negligible fluctuation at the "entrance" becomes huge at the "exit". At bifurcation points, the behavior of the system is unpredictable, and any randomness will move it to a new attractor. This is similar to the move in the chess game - after it appears a lot of options for the development of events.

To the right you will go - you will lose a horse ...

Crossroads in Russian fairy tales - this is a very bright image with a choice and uncertainty of the subsequent state of the system. With the approach to the bifurcation point, the system seems to oscillate, and the smallest fluctuation can lead to an entirely new organization, to order through fluctuation. And at this point in the crisis, it is impossible to predict the choice of the system. This is how in synergetics absolutely small causes give rise to huge consequences, revealing an unstable world of development of all systems - from the Universe to the selection of Buridan's ass.

Butterfly Effect

The arrival of the system to order through fluctuation, the formation of an unstable world, dependent on the slightest random changes, is reflected by the metaphor "butterfly effect". Meteorologist, mathematician and synergeticist Edward Lorenz (1917-2008) described the sensitivity of the system to the slightest changes. It belongs to him the image that one stroke of a butterfly's wing in Iowa can cause an avalanche of various processes that will end in Indonesia with a rainy season. The bright image was immediately picked up by writers, writing more than one novel on the subject of the multiplicity of developments. Popularization of knowledge in this field is largely due to Hollywood director Eric Bress with his cash movie "The Butterfly Effect."

Bifurcations and disasters

Bifurcations can be soft and hard. The feature of soft bifurcations is small differences in the system after passing through the bifurcation point. When the attractor has significant differences in the existence of the system, it is said that this bifurcation point is a catastrophe. For the first time such a concept was introduced by the French scientist René Federic Tom (1923-2002). He is the author of the theory of catastrophes, as bifurcation systems. Its seven elementary catastrophes carry very interesting names: a fold, an assembly, a tail of a swallow, a butterfly, a hyperbolic, an elliptical and a parabolic umbilic.

Applied synergetics

Synergetics and the theory of bifurcation are not so far from everyday life as it may seem. In the ordinary life, a person passes the point of bifurcation a hundred times during the day. The pendulum of our choice - conscious or only seeming conscious - swings constantly. And maybe understanding the processes of the synergetic organization of the world will help us make a more conscious choice, not reaching catastrophes, and bypassing small bifurcations.

Today all our knowledge of basic sciences has reached the point of bifurcation. The discovery of dark matter and the ability to save it put humanity at the point where an accidental change or discovery can lead us to a state that is difficult to predict. Modern study and mastery of outer space, the theory of "rabbit holes" and the space-time tube expand the possibilities of knowledge to unimaginable boundaries. It remains only to believe that by approaching the next bifurcation point, random fluctuation will not push mankind into the abyss of non-existence.

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