Integrabilnost in linearizabilnost persistentnih p : -q resonantnih polinomskih sistemov navadnih diferencialnih enačb : doktorska disertacija

slovenian

doktorska dela

2022

dissertations   first integral   focus quantities   integrability problem   linearizability quantities   p : -q resonant system   persistent resonant center   polynomial system of ODE's

Univerza v Mariboru, Fakulteta za naravoslovje in matematiko

One of the main problems in the theory of ordinary differential equations is the integrability problem, since it gives an important essential insight to theory of differential equations and the way to use differential equations in studies of dynamical processes in real-world systems. The main problems considered in this are the center problem and its closely related integrability and linearizability problems of p:-q resonant systems with quadratic nonlinearities. We also explain the relation between p:-q resonant system and the corresponding persistent p:-q resonant system in the sense of integrability and linearizability problems. In the first chapter, some basic notations of the commutative computational algebra are presented, mainly the properties of polynomial ideals and their varieties. We introduce the center and linearizability problems and describe the complexification of real systems. The second chapter is devoted to the generalization of the notion of a persistent center to a persistent p:-q resonant center. We find all conditions for existence of the persistent p:-q resonant center for several p:-q resonant systems with quadratic nonlinearities. We also prove the relation between integrability of p:-q resonant system and the corresponding persistent p:-q resonant system. In the third chapter, linearizability of 2:-3 resonant system with quadratic nonlinearities is studied. The conditions for linearizability are obtained by computing the ideal generated by the linearizability quantities and its decomposition into associate primes. In order to complete calculations we use an approach based on modular computations. The sufficiency of the obtained conditions is proven by several methods, mainly by the method of Darboux linearization. In the last chapter we define linearizability of a persistent p:-q resonant centers and unveil the link between linearizability transformation for p:-q resonant systems on one hand and linearizability transformation for persistent p:-q resonant systems on the other hand. Finally, the obtained results are demonstrated on persistent 1:-2 and 2:-3 resonant systems with quadratic nonlinearities.

08.08.2023