Theoretical background. A number of researches claim that the classical theory of dynamic systems ignores spe-cial cases of incomplete equivalence of mathematic transformations descriptions. Sometimes it is even argued that (con-trary to a prevailing paradigm) the study of purely discriminatory polynomial of control system (the system of differen-tial equations) fails to guarantee the correct judgments about the parametrical stability and system’s stability factors as the probably wrong interpretation of stability may result in accidents and even catastrophes caused by a defectively designed object. Such conclusion obviously ensues from the fact that there are examples of the systems that have the same discriminatory polynomial but differ substantially in the parametrical stability and stability factors under the vari-Bulletin of Lviv State University of Li fe Safety, №19, 2019 35 able parameters. These researches are concerned about the fact that generally used packages of applied programs – for they usually require the equivalent in the classical sense consolidation of differential equations system to a single “standard” form – are not able to secure the veracity of dynamic systems computation and to guarantee the correctness of their characteristics analysis without the application of additional controlling subprograms. For example, there may exist the risks of stability losses in the initial system, however being brought to the differential equations of first order, as a common practice, these risk will become absolutely imperceptible, and, as a result, the source of dangerous casual-ties may occur – accidents and catastrophes in case of the system material embodiment. Thus it is categorically declared the necessity of substantial researches in correctness of the results of engineers and IT specialists and of relevant amendments of bachelors and masters degrees curriculum. The purpose of the research. Thus, it is natural that there is a necessity to find out whether the previously imper-ceptible risks of accidents and catastrophes do exist and whether the classical dynamic systems theory does not take into consideration the unexpected possibilities of its problems correctness losses as a result (in the process) of their equiva-lent transformations. The aim of this article is to substantiate the essence and content of this kind “discoveries”. The paper provides a comprehensive analysis of the system’s simple examples that are to prove the possible risks from the equivalent, in classical sense, transformations of mathematical descriptions.Results and discussion. It has been found out that after the equivalent transformations instability as well as incor-rectness in fact do not “hide”, they do not become invisible and untraceable. The researchers rather consciously do not pay attention to the possible substantial deformations of the system. For indeed, in case of the reduction of the system description to the form of the normal system of differential equations of first order the possibilities of the stability loss become invisible not because the transformations were nonequivalent but because the variability of the system order is not prognosticated, and, therefore, the treatments of initial (where the change of order is obvious) and final systems differ considerably. Here at, the controller equation — the defined first integral — is the manifestation of one more possible system order which cannot be ignored. Actually, much depends on how we define, see, read, interpret the ana-lytical description of a certain phenomenon or process. Different characteristic determinants that identify, materially, different dynamic systems may correspond to the same characteristic polynomial. The determinant may be consciously equivalently transformed (deformed), and any transformed (deformed) determinant will identify a new system. Thus, any transformation – is, without exaggeration, the creation of something new, something different. The process of solving simple linear differential equations with fixed factor and their variation with the aim of so-lutions stability or analytical descriptions correctness evaluation is reduced to the solution of a relevant algebraic prob-lem and the research of its properties and characteristics. Consequently, there is no point in expecting any enigmatic or dramatic unexpectancies when the research is sophisticated and profound. Conclusions. The characteristic determinant reflects the properties of any system more deeply than the character-istic polynomial does. Any equivalent transformations of the system are always visible in the structure of the determi-nant, even if they are not defined in its equation roots (zeroes). In the result of equivalent transformations there certainly emerges a new formation – it looks like the same system but with new properties (otherwise there will be no necessity in any transformations). The loss of robustness is treated as an unexpectancy occurring as a result of motivated defor-mation of the system which is easy prognosticated. Nonrobust systems could have their own perspective. Their exten-sive application is advancing.
A Normal and Standard Form Analysis of the JWKB Asymptotic Matching Rule via the First Order Bessel’s Equation