| SINGULARLY PERTURBED SYSTEMS IN SUSTAINABLE MATHEMATICS APPLICATIONS (THEORETICAL AND APPLIED ASPECTS) Lyudmila Kuzmina1; 1KAZAN AVIATION INSTITUTE – KAZAN STATE UNIVERSITY, Kazan, Russian Federation; PAPER: 136/Mathematics/Regular (Oral) SCHEDULED: 15:55/Wed. 30 Nov. 2022/Arcadia 3 ABSTRACT: The paper is devoted to the different aspects of qualitative analysis in dynamics of complex nonlinear systems, that are generated by applied problems of engineering practice, including fundamental problems of modelling in mechanics. Main aims are the problems of optimal mechanical-mathematical modelling and the regular schemes of decomposition in engineering design. Multiconnectivity, high-dimensionality, nonlinearity of original statement under good detalization of full initial system lead to the necessity of the problem narrowing. The generalization of reduction principle, well-known in stability theory of A.M.Lyapunov, is important goal for engineering practice. Besides, the investigated objects are treated for unified view point on formed basic postulates (stability and singularity) as singularly perturbed ones (in sense of A.N.Tikhonov, A.Nayfeh, S.Cambell), with Sustainable Mathematics Applications. Uniform methodology, based on Lyapunov’s methods, in accordance with Chetayev’s stability postulate, is developed for mechanical systems with multiple time scales. The presented approach, with combination of stability theory and perturbations theory methods, allows to elaborate the general conception of the modelling, to build regular algorithm for constructing of the effective mechanical-mathematical models, to work out the simple schemes of engineering level for decomposition-reduction of original models and dynamic properties. This approach enables to obtain the simplified models, presenting interest for applications, with rigorous substantiation of the acceptability. The conditions of qualitative equivalence between full model and simplified models are determined. In the applications to mechanics (for mechanical systems with gyroscopes, for electromechanical systems, for robotic systems,…) the obtained results enable to construct the models (known and new ones) by strict methods, with the substantiation of the correctness for problems of analysis and synthesis. The interpretation of these models leads to new approximate theories, acceptable in applications of engineering practice. It allows to optimize the modelling process, to cut down the engineering design time. As applications, the different examples of concrete physical nature are considered. Besides the hierarchy of state variables is established by natural way automatically; the sequences of nonlinear shortened models (as comparison systems) are built in accordance with hierarchic structure of variables; the correspondence between original model and shortened one is revealed. The obtained results are generalizing and supplementing ones, known in theory of perturbations; these results are developing interesting applications in engineering. With reference to Mechanics the rigorous theoretic justification is obtained for considered approximate models and theories, both traditional (K.Magnus, A.Andronov, D.Merkin,…) and new ones (in particular, inertialess model, precessional model, Aristotel’s model of point mass dynamics,…). The author is grateful to Russian Foundation of Fundamental Investigations for support of this research. |
