Cycles in continuous and discrete dynamical systems : computations, computer-assisted proofs, and computer experiments

Abstract

The present work is devoted to calculation of periodic solutions and bifurcation research in quadratic systems, Lienard system, and non-unimodal one-dimensional discrete maps using modern computational capabilities and symbolic computing packages.In the first chapter the problem of Academician A.N. Kolmogorov on localization and modeling of cycles of quadratic systems is considered. For the investigation of small limit cycles (so-called local 16th Hilbert’s problem) the method of calculation of Lyapunov quantities (or Poincaré-Lyapunov constants) is used. To calculate symbolic expressions for the Lyapunov quantities the Lyapunov method to the case of non-analytical systems was generalized. Following the works of L.A. Cherkas and G.A. Leonov, the symbolic algorithms for transformation of quadratic systems to special Lienard systems were developed. For the first time general symbolic expressions of first four Lyapunov quantities for Lienard systems are obtained. The large limit cycles (or "normal" limit cycles) for quadratic and Lienard systems with parameters corresponding to the domain of existence of a large cycle, obtained by G.A. Leonov, are presented. Visualization on the plane of parameters of a Lienard system of the domain of parameters of quadratic systems with four limit cycles, obtained by S.L. Shi, is realized.The second chapter of the thesis is devoted to non-unimodal one-dimensional discrete maps describing operation of digital phase locked loop (PLL). Qualitative analysis of PLL equations helps one to determine necessary system operating conditions (which, for example, include phase synchronization and clock-skew elimination). In this work, application of the qualitative theory of dynamical systems, special analytical methods, and modern mathematical packages has helped us to advance considerably in calculation of bifurcation values and to define numerically fourteen bifurcation values of the DPLL’s parameter. It is shown that for the obtained bifurcation values of a non-unimodal map, the effect of convergence similar to Feigenbaum’s effect is observed

Elena Kudryashovan väitöstutkimuksessa selvitettiin matemaattisten mallien ratkeavuutta käyttäen nykyaikaisia laskentamahdollisuuksia ja symbolisen laskennan ohjelmistoja. Tutkimuksen käytännön sovellutukset liittyvät tietokoneiden toiminnan tehostamiseen.