| Dynamical systems are important in many fields of science and technology includingphysics, engineering, life science, and etc. It is therefore necessary todevelop efficient and accurate numerical approaches to simulate various complexsystems. This dissertation focuses on Eulerian approaches for computational dynamicalsystems based on the level set method. Based on the theory of ergodicpartition, we have first developed a concept called coherent ergodic partitionwhich can be used as a tool for quantifying the level of mixing. Numerically,we have also developed an efficient Eulerian approach to extract such invariantset in the extended phase space. Applying some recently developed Eulerianalgorithms for long time flow map computations, we then propose a new partialdifferential equation (PDE) approach for measuring the chaotic mixing propertyof a dynamical system. We introduce a numerical quantity named VIALS whichdetermines the temporal variation of the averaged surface area over all level surfacesof an advected function. Finally, we propose a new variational approachfor extracting limit cycles in dynamical systems. The minimization process canbe efficiently carried out by converting the functional to the Rudin-Osher-Fatemi(ROF) model for image regularization. |
