VIALS: An Eulerian tool based on total variation and the level set method for studying dynamical systems

We propose a new Eulerian tool to study complicated dynamical systems based on the average growth in the surface area of a family of level surfaces represented implicitly by a level set function. Since this proposed quantity determines the temporal variation of the averaged surface area of all level surfaces, we name the quantity the Variation of the Integral over Area of Level Surfaces (VIALS). Numerically, all these infinitely many level surfaces are advected according to the given dynamics by solving one single linear advection equation. To develop a computationally efficient approach, we apply the coarea formula and rewrite the surface area integral as a simple integral relating the total variation (TV) of the level set function. The proposed method can be easily incorporated with a recent Eulerian algorithm for efficient computation of flow maps to speed up our approach. We will also prove that the proposed VIALS is closely related to the computation of the finite time Lyapunov exponent (FTLE) in the Lagrangian coherent structure (LCS) extraction. This connects our proposed Eulerian approach to widely used Lagrangian techniques for understanding complicated dynamical systems.