Stability Theory Methods In Singularly Perturbed Systems Dynamics
by
Lyudmila K. Kuzmina
Kazan Aviation Institute, Adamuck, 4-6, Kazan-15, 420015, Russia

The work is get involved to the different problems of qualitative analysis in general theory of dynamic systems with applications to mechanics problems, with the development of the reduction principle in general analysis. Non-linearity, high dimensionality, multi-connectivity are causing the impediments for the obtaining exact solution by analytical (and analytic-computer) methods. It leads to the necessity of the decomposing original system, with the separation of systems parameters on substantial and non-substantial ones, with the revealing of freedom degrees hierarchy, with the subsequent transition to the shortened subsystem. These problems are important both for general theory of dynamic systems and for applications. With reference to a mechanics problems, to the special peculiarities of mechanical systems, it leads to the singularly perturbed problems of dynamic systems, with the different singularities types, with critical cases, with the non-linear singular generating systems. The principal questions are discussed in research: the methodology of the building optimal comparison systems; the development of the decomposition rigorous manners; the substantiation of legitimacy of shortened subsystems in dynamics; the determination of the qualitative equivalence conditions. The work is formed on the accepted basic assumptions, that are ascending to the Chetayev’s stability statements and to the Kuzmin’s parametric stability property. The employed method is developing the uniform approach, based on stability postulate and singularity postulate, that is combining Lyapunov’s methods with the perturbations theory methods. Besides the state of original object may be described by the mathematical model with the singular perturbations; the shortened model is corresponding to the singularly perturbed system, that is asymptotic approximation for original one. The constructiveness of this approach is illustrated on concrete examples, with reference to the applications in Mechanics.

The author thanks to Russian Foundation of Fundamental Investigations for research support.

Date received: October 14, 2002