Opis: The Moon-Rand systems, developed to model control of flexible space structures, are systems of differential equations on ▫$RR^3$▫ with polynomial or rational right hand sides that have an isolated singularity at the origin at which the linear part has one negative and one pair of purely imaginary eigenvalues for all choices of the parameters. We give a complete stability analysis of the flow restricted to a neighborhood of the origin in any center manifold of the Moon-Rand systems, solve the center problem on the center manifold, and find sharp bounds on the number of limit cycles that can be made to bifurcate from the singularity when it is a focus. We generalize the Moon-Rand systems in a natural way, solve the center problem in several cases, and provide sufficient conditions for the existence of a center, which we conjecture to be necessary. Ključne besede:matematika, cikličnost, integrabilnost, stabilnost, mathematics, cyclicity, integrability, stability Objavljeno: 10.07.2015; Ogledov: 142; Prenosov: 0 Polno besedilo (0,00 KB)

Opis: Using methods of computational algebra we obtain an upper bound for the cyclicity of a family of cubic systems. To that end we overcome the problem of nonradicality of the associated Bautin ideal by moving from the ring of polynomials to a coordinate ring. Finally, we also determine the number of limit cycles bifurcating from each component of the center variety. Ključne besede:matematika, cikličnost, limitni cikli, računalniška algebra, mathematics, cyclicity, limit cycles, computer algebra system Objavljeno: 10.07.2015; Ogledov: 121; Prenosov: 0 Polno besedilo (0,00 KB)

Opis: This doctoral dissertation is devoted to the studies of some qualitative properties of certain polynomial systems of ordinary differential equations. The main problems that are considered in this thesis are the problems of integrability and cyclicity. Some results on the classification of the global phase portraits of a family of cubic systems are presented as well. In the first chapter basic notions and results of the qualitative theory of systems of ODE's are introduced. Since one of important tools for our study of these problems is the commutative computational algebra, some main notions and properties of polynomial ideals and their varieties, including various algorithms related to them, are also presented in the introduction. In the second chapter methods for investigation of trajectories near degenerated singularities are presented. They are further used for the classification of global phase portraits of a family of cubic systems with the nilpotent center at the origin. In the third chapter the main problem of these thesis is studied, the problem of integrability. The problem of integrability which is connected to the problem of distinguishing between a center and a focus is studied for two different families of cubic polynomial systems of ODE's. With the computational algebra approach the necessary conditions for the existence of the first integral of these systems were obtained. For all but one condition was proven, using various approaches, the existence of the first integrals. The center problem for the real systems can be generalized to the complex systems. The origin of the system obtained after the complexification of the real system is the so-called 1:-1 resonant singular point, from which one additional generalization follows. This is the generalization to the p:-q resonant center. In the third chapter the :-3 resonant singular point of a quadratic family of complex systems is studied. The fourth chapter is devoted to the study of the problem of integrability of a three dimensional polynomial system with quadratic nonlinearities. The problem of existence of two independent first integrals and the existence of one first integral in the system was investigated. In the last chapter local bifurcations of limit cycles of a family of cubic systems are studied. Estimations for the number of limit cycles bifurcated from each components of the center variety are obtained. Ključne besede:planar systems of ODE's, higher dimensional systems of ODE's, phase portrait, nilpotent center, limit cylces, Poincaré compactification, center problem, Bautin ideal, focus quantities, time-reversibility, integrability problem, Darboux method, linearizability, limit cycle, cyclicity Objavljeno: 19.07.2016; Ogledov: 361; Prenosov: 28 Polno besedilo (12,26 MB)