Year: 2012
Type: dissertation
Abstract: Vagaytsev, Vladimir Analytical-numerical methods for finding hidden oscillations in dynamical systems Jyvaskyla: University of Jyvaskyla, 2012, 40 p.(+included articles) (Jyvaskyla Studies in Computing ISSN 1456-5390; 158) ISBN 978-951-39-4984-6 (nid.) ISBN 978-951-39-4985-3 (PDF) Finnish summary Diss. This work is devoted to the investigation of oscillations in multi-dimensional nonlinear dynamical systems. Localization of classical self-excited oscillations doesn’t provide any computational complexity. Such oscillations can be easily computed by means of standard computational procedure: one should determine equilibria of the system under consideration and build the trajectory of the system with initial data from a point in a neighborhood of an unstable equilibrium using standard numerical methods. Besides the self-excited oscillations there exist “hidden oscillations”, the existence of which is not obvious. Arising in electrical circuits, phase-locked loops, control systems and another complex dynamical systems, such oscillations can lead to malfunction. Usually, oscillations of this type can not be localized either by the standard numerical computational approach described above or by pure analytical methods. From the computational point of view, arises an interesting and important problem: “what” should be computed and “where” shall we do it? An answer to this question can be obtained by a synthesis of analytical and numerical methods. In this thesis a new effective analytical-numerical method of investigation of oscillations in multi-dimensional dynamical systems is suggested. This method is applied to classical Chua’s circuits with 5 linear elements and one piecewiselinear element which is called “Chua’s diode”. As a result, a hidden attractor in Chua’s circuit is obtained for the first time. Suggested method is also applied to various modifications of Chua’s system: smooth Chua’s system (with hyperbolic tangent nonlinearity) and discontinuous Chua’s system (with signum nonlinearity). As a result, hidden attractors in modified Chua’s systems are obtained. Detailed description and justification of the method mentioned above and numerical modeling results are presented in the included articles. MATLAB implementation of the suggested analytical-numerical method is presented in appendices.
Author: Vladimir Vagaytsev
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2336478895