1. Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loopFeng Liang, Maoan Han, Valery Romanovski, 2012, izvirni znanstveni članek Opis: In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with the center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclinic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in ▫$epsilon$▫. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function. Ključne besede: mathematics, limit cycle, homoclinic loop, bifurcation Objavljeno: 07.06.2012; Ogledov: 839; Prenosov: 10 Celotno besedilo (0,00 KB) |
2. An integrity analysis of washing machine holdersNenad Gubeljak, Jožef Predan, Matej Mejač, 2007, izvirni znanstveni članek Opis: The paper deals with structure integrity analysis of the holder for the carrying cross of a washing machine drum. Premature fracture of the holder occurred between mechanical sustainable tests of washing machine in the factory. In order to prevent fracture, the task was to determine the reasons for premature fracture of the holder and to estimate the suitability of the new design of the holder cross. Input data for structure integriry analysis were obtained by material by mechanical testing of used materials. Stress and strain analysis of holder limit load was performed by finite element modeling of holder. Dynamic tests of holders with two different thicknesses were done on a servo-hydraulic machine in order to find dynamical strength and endurance of holder. Fracture behaviour of holders is defined as initiation and propagation of crack. The obtained behaviour confirmed that the new design of holders reduces stress concentration in the critical region. Consequently, the new holder subjected to the same dynamic load can endure a higher number of cycles until breakage. The total number of cycles overcomes industrial testing requirements. Ključne besede: lomna mehanika, visokociklično utrujanje, preizkušanje lomne žilavosti, nosilci kadi pralnega stroja, mejna obremenitev, ocena celovitosti konstrukcije, fracture mechanics, high cycle fatigue, fracture toughness testing, washing machine holders, washing machine drums, structure integrity assessment, limit load Objavljeno: 10.07.2015; Ogledov: 445; Prenosov: 25 Celotno besedilo (517,04 KB) |
3. Qualitative Studies of Some Polynomial Systems of Ordinary Differential EquationsMaša Dukarić, 2016, doktorska disertacija 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: 586; Prenosov: 63 Celotno besedilo (12,26 MB) |
4. Limit cycle bifurcated from a center in a three dimensional systemBo Sang, Brigita Ferčec, Qin-Long Wang, 2016, izvirni znanstveni članek Opis: Based on the pseudo-division algorithm, we introduce a method for computing focal values of a class of 3-dimensional autonomous systems. Using the $Є^1$-order focal values computation, we determine the number of limit cycles bifurcating from each component of the center variety (obtained by Mahdi et al). It is shown that at most four limit cycles can be bifurcated from the center with identical quadratic perturbations and that the bound is sharp. Ključne besede: algorithms, three dimensional systems, focal value, limit cycle, Hopf bifurcation, center Objavljeno: 08.08.2017; Ogledov: 288; Prenosov: 38 Celotno besedilo (236,33 KB) |