Τοπολογική ταξινόμηση δυναμικών συστημάτων

Abstract

The topological classification and study of vector fields is the subject of this thesis. In Chapter 1 the necessary definitions are given, along with the known results on the classification of vector fields on 1-dimensional and 2-dimensional manifolds. In Chapter 2 methods of Knot Theory are used for the clarification of the topological study of some strange attractors found in the bibliography. In Chapter 3 a technique is developed, which can be used to classify globally vector fields defined on Euclidean spaces of any dimension. This technique is then used to classify some vector fields of R^2 and R^3. In the final Chapter 4 a vector field of R^3 is studied which is invariant under the D_2 symmetry group. We present its global phase portrait, for various parameter values, and its partial bifurcation diagram. The existence of chaos is proven and its connection to the symmetry properties of the attractor is discussed. We end its study presenting its behavior at infinity.