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- ItemParameterizations of sub-attractors in hyperbolic balance laws(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Ehrt, JuliaThis article investigates the properties of the global attractor of hyperbolic balance laws on the circle, given by : u_t+f(u)_x=g(u). The new tool of sub-attractors is introduced. They contain all solutions on the global attractor up to a given number of zeros. The article proves finite dimensionality of all sub-attractors, provides a full parameterization of all sub-attractors and derives a system of ODEs for the embedding parameters that describes the full PDE dynamics on the sub-attractor
- ItemCascades of heteroclinic connections in hyperbolic balance laws(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Ehrt, JuliaThe 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.
- ItemSlow motion of quasi-stationary multi-pulse solutions by semistrong interaction in reaction-diffusion systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Wolfrum, Matthias; Ehrt, JuliaIn this paper, we study a class of singularly perturbed reaction-diffusion systems, which exhibit under certain conditions slowly varying multi-pulse solutions. This class contains among others the Gray-Scott and several versions of the Gierer-Meinhardt model. We first use a classical singular perturbation approach for the stationary problem and determine in this way a manifold of quasi-stationary $N$-pulse solutions. Then, in the context of the time-dependent problem, we derive an equation for the leading order approximation of the slow motion along this manifold. We apply this technique to study 1-pulse and 2-pulse solutions for classical and modified Gierer-Meinhardt system. In particular, we are able to treat different types of boundary conditions, calculate folds of the slow manifold, leading to slow-fast motion, and to identify symmetry breaking singularities in the manifold of 2-pulse solutions.