The Dissertation investigates the relation between global attractors of hyperbolic balance laws and viscous balance laws on the circle. Hence it is thematically located at the crossroads of hyperbolic and parabolic partial differential equations with one-dimensional space variable and periodic boundary conditions given by: (H): u_t + [f(u)]_x = g(u) and (P): u_t + [f(u)]_x = e u_xx + g(u). The results of the work can be split into two areas: The description of the global attractor of equation (H) and the persistence of solutions on the global attractor of (P) when e vanishes. The key idea of the work is the introduction of finite dimensional sub-attractors. This tool allows to overcome several difficulties in the description of the global attractor of equation (H) and closes one of the last remaining gaps in its complete description: Theorem 2.6.1 yields a complete parameterization of all finite dimensional sub-attractors in the hyperbolic setting. The second main result corrects a result on the persistence of heteroclinic connections by Fan and Hale [FH95] for the case e-->0 (Connection Lemma 3.2.8). The Cascading Theorem 3.2.9 then yields convergence of heteroclinic connections to a cascade of heteroclinics in case of non-persistence. In addition to the introduction and conclusions, the work consists of three chapters: Chapter 2 gives a self contained overview about what is known for global attractors for both equations and concludes with the result on the parameterizations of the sub-attractors of the hyperbolic equation (H). Chapter 3 is exclusively concerned with the question of persistence. The two main results on persistence (the Connection Lemma and the Cascading Theorem) are stated and proved. Chapter 4 concludes with geometrical investigations of persisting and non-persisting heteroclinic connections for e-->0 for some low dimensional sub-attractor cases. Not all results are rigorous in this chapter.