In this thesis, we develop a simple numerical algorithm to estimate the Finite Time Lyapunov Exponent (FTLE) for extracting the Lagrangian coherent structures (LCS) in dynamical systems from a sparse set of limited Lagrangian particles' trajectories. Our proposed method is based on the reconstruction of the flow field, which is accomplished by radial basis function (RBF). After reconstructing the flow field, the flow map of the dynamical system is determined by any typical methods. Other than this simple algorithm, we also develop an algorithm based on the Schur complement for updating, rather than recomputing, the reconstruction in RBF when new trajectory data is made available in applications. In addition, we impose impenetrable boundary condition to the reconstructed flow field in th...[ Read more ]

In this thesis, we develop a simple numerical algorithm to estimate the Finite Time Lyapunov Exponent (FTLE) for extracting the Lagrangian coherent structures (LCS) in dynamical systems from a sparse set of limited Lagrangian particles' trajectories. Our proposed method is based on the reconstruction of the flow field, which is accomplished by radial basis function (RBF). After reconstructing the flow field, the flow map of the dynamical system is determined by any typical methods. Other than this simple algorithm, we also develop an algorithm based on the Schur complement for updating, rather than recomputing, the reconstruction in RBF when new trajectory data is made available in applications. In addition, we impose impenetrable boundary condition to the reconstructed flow field in the case that an obstacle exists in the computational domain. The effectiveness of the proposed method will be demonstrated by some examples including autonomous flows, periodic flows, and also measurement from real-life data.