Realization of the Iteration Procedure in Localization Problems of Autonomous Systems

• Main Authors: A. N. Kanatnikov, A. P. Krishchenko
• Format: Article
• Language: Russian
• Published: MGTU im. N.È. Baumana 2014-01-01
• Series: Nauka i Obrazovanie
• Subjects: dynamical system; localization; localizing set; invariant compact set; iteration procedure
• Online Access: http://technomag.edu.ru/jour/article/view/738

<p>In the last 15 years one way for a qualitative analysis of dynamical systems was formed i.e. the localization of invariant compact sets of a dynamical system. Here the localization means creating a system of such sets, which contain all invariant compact sets of a dynamic system [1], in the phase space.</p><p>Invariant compact sets are closely connected with bounded trajectories of the system, the structure of which in the phase space play key role in many applications of dynamical system theory. The problems of invariant compact sets localization abut upon other important problems, for instance, the problems of estimation of attractor basins, control problems, etc.</p><p>Back investigations of localization problems was oriented both to development of solving methods [28] and to investigation of particular dynamical systems encountered in applications (see, for example, [9 { 16]).</p><p>One of quite efficient methods of localization problem solving is based on smooth functions defined in the phase space. It is so called functional method [1 { 3]. Effectiveness of the method is enhanced when we use several functions. Thus, using the next function gives the restriction of the already constructed localizing set. An iteration procedure for sequential narrowing of the localizing set [1 { 2] arises.</p><p>The paper presents analysis of the iteration procedure, which naturally occur in the autonomous systems of special type where the right side of each differential equation is resolvable relative to the corresponding phase variable. Such systems are encountered in applications [17].</p>