School of Science Department of Mathematics 51 Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG Shing Yu / MATH Student: XU Zian / DSCT Course: UROP1100, Fall In the original approach, we partition different initial conditions based on the positions of particles at a fixed final time tN as in finite-time Lyapunov exponent (FTLE). However, two trajectories may have a large FTLE at some time between t0 and tN, even if they have a small FTLE at the final time tN. To address this problem, we want to partition different initial conditions of particles based on the whole trajectories from the initial time t0 to the final time tN. And we try to do this by clustering the matrix formed by the positions at different times between t0 and tN using k-means. Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG Shing Yu / MATH Student: YAU Wing Yan / MATH-AM Course: UROP2100, Fall We investigate a quantity to describe the flow motion, in the sense that how the infinitesimal portion (area for 2d flow, volume for 3d flow) changes over time, as well as testing the performance of two different Lagrangian approaches, that one of them approximate the final positions of grid points to obtain the Jacobian matrix, while another one keeps track on the deformation of infinitesimal portion to do so. Numerical examples of compressible and incompressible flows are presented, and their expected behaviour and realcase behaviour are discussed. Efficient Algorithms for Visualizing Dynamical Systems Supervisor: LEUNG Shing Yu / MATH Student: YIP Kwun Chuen / MAEC Course: UROP1100, Fall The Finite Time Lyapunov Exponent (FTLE) is a numerical tool for calculating the rate of separation of close initial values in a dynamic system. The report shows an implementation of the FTLE into macroeconomic fields to find a suitable domain of parameters for an economy that implements an integrable feedback expenditure policy in a financial crisis. The report will cover the model and the economic intuition behind it. The variable mainly used is the speed of response of government targeted demand towards the actual demand of the economy and the magnitude of the integrable policy. With these two variables, the report shows the output level and output growth of the economy within a decade of the policy implementation.