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## Scientific Bulletin. Physical and Mathematical Research

#### Abstract

The objective of the theory of stability of motion is to establish signs that make it possible to judge whether the motion in question is stable or unstable. Since in reality perturbing factors always inevitably exist, it becomes clear that the problem of stability of movement assumes very important theoretical and practical significance.

Mathematical modeling of processes and phenomena in animate and inanimate nature always involves a certain classification of them in accordance with their complexity. Many processes and phenomena are modeled by large-scale systems (CMS), which consist of separate subsystems, united by communication functions. In many cases, CMS is characterized by certain signs described by A.I. Kuhtenko. Among them: multidimensionality; variety of system structure (networks, hierarchical structures, etc.); the multiplicity of elements of the system (the relationship of sub-systems at one level and communication at different levels of the hierarchy); the diversity of the nature of elements (machines, machine- robots, etc.); the multiplicity of changes in the composition and state of the system (variability in the structure, connections and composition of the system) multi-criteria system (the difference between local criteria for subsystems and global criteria for the system as a whole, their inconsistency), etc. Systems with such properties in most practically important cases are difficult to study for the dynamic properties of their solutions.

The purpose of the study of this work is the development of stability criteria (asymptotic stability) and instability of equilibrium states of linear and nonlinear large-scale systems, application of the results to the stability problems of the isodromic regulator.

In this paper, we describe an approach to the study of the stability of the equilibrium state of nonlinear QMS using some auxiliary linear systems. This approach is based on a special representation of scalar or vector functions through strictly positive functions.

Sufficient conditions for stability, asymptotic stability, and instability of the equilibrium state of the system under consideration are obtained. In the end, we obtain sufficient conditions for the asymptotic stability of the system, the rigidity and damping of which are nonlinear and depend explicitly on time.

#### First Page

51

#### Last Page

61

#### References

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9 Martynyuk, A.A., Mullajonov, R.V. (2010). Ob odnom metode issledovaniya ustojchivosti nelinejnyh krupnomasshtabnyh sistem [On a method for studying the stability of nonlinear large-scale systems]. Prikladnaya mekhanika. 5(46). Pp. 125-134.

10. Mullajonov, R.V. (2016). Analiz ustojchivosti nekotoryh krupnomasshtabnyh sistem [Stability analysis of some large-scale systems]. Tashkent: Yangi asr avlodi.

#### Recommended Citation

Mullajonov, Rustamjon V.; Abdugapparova, Shakhodathon N.; and Mirzaahmedova, Jumagul V.
(2020)
"ON THE STABILITY OF SOME NON-STATIONARY NONLINEAR SYSTEMS,"
*Scientific Bulletin. Physical and Mathematical Research*: Vol. 2
:
Iss.
1
, Article 7.

Available at:
https://uzjournals.edu.uz/adu/vol2/iss1/7