Integrabilnost, linearizabilnost in limitni cikli polinomskih sistemov navadnih diferencialnih enačb : doktorska disertacija

slovenian

doktorska dela

2023

dissertations   Hamiltonian systems   integrability   limit cycles   linearizability   reversibility   systems of ordinary differential equations

Univerza v Mariboru, Fakulteta za naravoslovje in matematiko

The main topic of this doctoral disertation is the qualitative analysis of some families of systems of ordinary differential equations (ODE). We study integrability, linearizability and limit cycles in two- and three-dimensional polynomial systems.In the introduction we define fundamental concepts connected to singular points of systems of ODE and behavior of trajectories in their neigbourhoods. We describe some main methods and algorithms of commutative computational algebra which are essential for the performed analysis of the systems.In the second chapter we define two important properties of n-dimensional systems of ODE, namely integrability and linearizability. First, we introduce a method to get conditions for integrability of a system and then we describe some techniques to prove the sufficiency of such conditions. In order to study linearizability we make a quick introduction into theory of normal forms. We explain a method to find conditions for linearizability of a system and prove an important theorem about linearizability of integrable systems. Using the theory mentioned above we study integrability and linearizability of a quadratic three-dimensional system with (1:-1:-1)-resonant singularity at the origin.The third chapter is devoted to planar systems of ODE and their linearizability, which is very closely related to isochronicity. We explain a method to get the conditions for linearizability of a system which works in some cases when we cannot get them from linearizability quantities. We focus on Hamiltonian systems with homogeneous and nonhomogeneous nonlinearities of degree at most seven. In the fourth part we address the center-focus problem for some reversible cubic systems. We analyse three two-dimensional systems which are transformed into one of the canonical forms of cubic systems with the singularity of center or focus type at the origin. We prove that all the obtained systems are Darboux integrable. At last, we study the orbital reversibility of the systems.In the last chapter we concentrate on limit cycles. We recall one of the key phenomena, responsible for emergence of limit cycles, Hopf bifurcation. We describe a method for investigation of the points at the infinity, Poincar\e compactification and an approach to analyse a neighborhood of degenerate singular points - directional blow up. Then we study the possibilities for existence of limit cycles in a three-dimensional biochemical model and determine the phase portrait in the first quadrant of a two-dimensional reaction model.

08.08.2023