StatusThe thesis was presented on the 31 March, 2016
Approved by NCAA on the 3 June, 2016
Abstract– 0.53 Mb / in romanian
– 0.53 Mb / in english
1.57 Mb /
The language of the Thesis is English. It comprises 154 base pages and has the following structure: Introduction, 4 Chapters, General Conclusions and Recommendations, Bibliography with 140 References. Research outcomes were reflected in 19 scientific publications. Keywords: cubic differential system, affine invariant polynomial, invariant straight line, multiplicity of a line, configuration of invariant straight lines, perturbed system.
Field of study: Qualitative Theory of Dynamical Systems, Invariant Theory of Differential Equations.
The purpose and objectives: to give a full classification for the family of cubic systems with invariant straight lines of total multiplicity eight; this classifications supposes the de- tection of all possible configurations of invariant lines for this family and the construction of affine invariant criteria for the realization of each one of the detected configurations.
Novelty and scientific originality. In our Thesis for the first time there are constructed all the possible configurations of invariant lines of total multiplicity eight for cubic systems. Our set of configurations contains as particular cases all the configurations detected by other authors in special cases. Additionally we give necessary and sufficient conditions for the realization of each one of the corresponding configurations. Moreover we completed the classification of Llibre and Vulpe detecting a new class of cubic systems with invariant lines of total multiplicity nine.
The main scientific problem which is solved in this Thesis consistsin classifying the whole family of cubic differential systems possessing invariant lines of total multiplicity eight according to configurations of these lines; this classification is very helpful for obtaining the complete topological classification of this family and is useful for the study of integrability of these systems.
The significance of theoretical and practical values of the work. The obtained in this thesis results concerning cubic systems with invariant lines of total multiplicity eight represent an important step in algebraic and geometric studies of cubic differential systems.
Implementation of the scientific results. They could be applied: (i) as a basis for
determining of the first integrals of such systems; (ii) for further investigations of cubic
systems with invariant lines of total multiplicity less then 8; (iii) in the study of some
mathematical models which describe processes in physics, chemistry, medicine and so on;
(iv) as a support for teaching courses in higher education.