Dynamical systems are important in many fields of science and technology including physics, engineering, life science, and etc. It is therefore necessary to develop efficient and accurate numerical approaches to simulate various complex systems. This dissertation focuses on Eulerian approaches for computational dynamical systems based on the level set method. Based on the theory of ergodic partition, we have first developed a concept called coherent ergodic partition which can be used as a tool for quantifying the level of mixing. Numerically, we have also developed an efficient Eulerian approach to extract such invariant set in the extended phase space. Applying some recently developed Eulerian algorithms for long time flow map computations, we then propose a new partial diff...[ Read more ]

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