In this master thesis we defined the basic definition and concepts of the dynamics systems in the plane. The dynamic of linear mapping in two and three dimensions depends on eigenvalues of their corresponding matrix representation, as well as the invariant subspaces. Then we defined Smale's horseshoe map and the associated symbolic dynamics. We showed that Smale's horseshoe map is reduced to its invariant subset and topologically conjugated to the shift map of the relevant itinerary space and we also explored some interesting properties. From the plane we moved to the torus to show that the map on the torus is chaotic, although their dynamics can be easily described. Then we looked at the third type of multidimensional dynamical phenomenon, which are the attractors. We observed two types of attractors, a solenoid and Plykin's attractor, as well as mappings on them which we have shown that to be chaotic. An important feature of all previous mappings is hyperbolicity and stable and unstable subspaces resulting from this feature. The most important result of this work is the Teorem [5.7] that tells us about the conditions of existence of stable and unstable manifolds.