One very important concept in understanding a dynamical system is coherent structure. Such structure segments the domain into different regions with similar behavior according to a quantity. When we try to partition space-time into regions according to a Lagrangian quantity advected along with passive tracers, such class of coherent structure is called the Lagrangian coherent structures (LCSs). Among many, a simple definition of an LCS uses the finite-time Lyapunov exponent (FTLE). It measures the rate of separation between adjacent particles over a finite time interval with an infinitesimal perturbation in the initial location. In the talk, we first present various Eulerian-based numerical methods which efficiently compute the flow maps of any continuous dynamical system and, therefore, the corresponding FTLE. Based on these techniques we developed, we will also propose some other useful numerical tools for extracting important structures hidden in the system.

# Applied Math Seminar: Eulerian Approaches based on the Level Set Method for Visualizing Continuous Dynamical Systems (Shingyu Leung, Department of Mathematics, HKUST)

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