Science and Education in Karakalpakstan issn 2181-9203 Science Magazine chief editor

Science and Education in Karakalpakstan. 2023 №2/1 ISSN 2181-9203 
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Let's give one more example to understand this situation. Let's start by holding two ends of a 
flexible stick. As the force increases, the rod will bend more and more. But when it reaches a 
certain stage, the stick breaks and it no longer has the properties of a stick. It is difficult to predict 
where and how it will break. In the same way, the uncertainty of the future is reflected in the 
peculiarity of the mechanism of bifurcation. 
At the bifurcation point, selection occurs in the system, and due to the presence of an 
element of chance, there is no possibility of predicting the selection of the trajectory of the system's 
evolution. Without the mathematical theory of bifurcations and breakdowns, it is practically 
impossible to understand and control the dynamics of complex nonlinear systems. Random changes 
in the components of complex dynamic systems show fluctuations in it. The combination of any or 
some of these fluctuations in the system is observed to increase as a result of feedback and leads to 
the disturbance of the previous state of the system. At the moment of discontinuity (bifurcation 
point) in the system, the random effect pushes the system to a new development path, and after the 
selection of one of the possible paths, one-valued determinism affects the development trajectory - 
this is the possibility of predicting the development of the system until the next bifurcation point. I. 
According to Prigogine, bifurcation processes indicate the complexity of the system. N. Moiseev 
states that every state of the social system is a state of bifurcation [4]. 
So, in complex nonlinear processes, chance and necessity complement each other. 
Approaching the bifurcation point under non-equilibrium conditions, the system is very 
sensitive to external influences, and even a small external influence can have an unexpected effect. 
Therefore, sometimes even very small fluctuations in states far from the equilibrium state can have 
a strong effect on the system, completely destroy the previous state of the system, and transform the 
system into another state. In essence, the theory of catastrophes is close and similar to the idea of 
self-organizing criticality (P.Bak, K.Chen). According to this, the interaction of a large number of 
elements in the system can lead to spontaneous (spontaneous) evolution of the system to a critical 
state, and even small effects can lead to destruction. Such complex systems include many natural 
and social systems. 
As the system becomes more complex, its size increases and the bifurcation (collapse) state 
increases. Consequently, as the complexity of the system increases, the possible ways of its further 
development, i.e. divergence, also increase. But the probability of two systems developing in 
exactly one channel (roads) is equal to zero. This means that the process of self-organization means 
that the number of forms of organization is increasing. 
This situation can be understood in the following example. Suppose two identical circular 
columns are under the same vertical pressure. They are affected by the incessant wind. Since the 
mechanical properties and vertical pressures of the columns are the same, they have exactly the 
same limit of stability. According to the theory of L. Euler, bifurcation (destruction) should occur 
simultaneously in both columns. However, since the wind speed is never exactly the same, the 
positions of the two columns are different after the bifurcation. As a result, in the new conditions, 
the vibration of the pillars takes place in different ways of evolution. In other words, they vibrate in 
different planes. Because there are too many equilibrium forms. 
It is known that these conditions are observed in self-organized processes, that is, "the 
system loses its stability due to the occurrence of a strong imbalance in the system. The parameters 
describing such a state are called critical, and are passed from this critical state to one of the 
possible new stable states by jumping. Such a branching possibility of the system development path 
is called a bifurcation point. The transition to any of the possible states is a matter of chance: at the 
bifurcation point, a large number of fluctuations occur, one of which by chance leads the system to 
a new stable state» [6]. 

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