The influence of small perturbation on phenomenon of delayed loss of sta-bility
DOI:
https://doi.org/10.52754/16948645_2025_4(1)_41Keywords:
small parameter; limit transition; eigenvalues; stability of solutions; integral curvesAbstract
The study of solutions to singularly perturbed problems remains relevant, as many mathematical models in technical and natural sciences are described by such differential equations. Despite existing research, there remains a need for a more in-depth analysis and further study of the influence of small perturbations on the phenomenon of delayed loss of stability. The aim of this study was to examine the influence of a small perturbation on the phenomenon of delayed loss of stability, as well as to justify the limit transition confirming the convergence of solutions of the perturbed and unperturbed problems. To achieve this goal, analytical methods were employed, including the level lines method and methods for selecting descending integration paths, which made it possible to rigorously substantiate the limit transitions between the perturbed and unperturbed problems. The study established that in the absence of a small perturbation, the phenomenon of delayed loss of stability persists regardless of the location of the zeros of the eigenvalues whether on the real axis or in the complex plane. In the presence of a small perturbation, the situation changes: if the eigenvalues have zeros on the real axis, the delay phenomenon does not occur. However, if the zeros are located in the complex plane, the delay is observed only over a limited time interval. In the case where the eigenvalues have poles, the small perturbation does not affect the presence of the phenomenon persists in all cases. Thus, the influence of a small perturbation on the delayed loss of stability depends significantly on the nature of the eigenvalues. It was also substantiated that, under certain conditions on the small perturbation, convergence of solutions is ensured when transitioning from the perturbed problem to the unperturbed one. The results of the study provide a justification for the existence and nature of delayed loss of stability in broader functional spaces, which is important for applied problems in modelling unstable processes
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