Computing Finite-Time Lyapunov Exponents with Optimally Time Dependent Reduction

We present a method to compute Finite-Time Lyapunov Exponents (FTLE) of a dynamical system using Optimally Time-Dependent (OTD) reduction recently introduced by H. Babaee and T.P. Sapsis (\emph{A minimization principle for the description of modes associated with finite-time instabilities}, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 472, 2016). The OTD modes are a set of finite-dimensional, time-dependent, orthonormal basis $\{ \mathbf{u}_i(\mathbf{x},t)\}|_{i=1}^N$ that capture the directions associated with transient instabilities. The evolution equation of the OTD modes is derived from a minimization principle that optimally approximates the most unstable directions over finite times. To compute the FTLE, we evolve a single OTD mode along with the nonlinear dynamics. We approximate the FTLE from the reduced system obtained from projecting the instantaneous linearized dynamics onto the OTD mode. This results in a significant reduction in the computational cost compared to conventional methods for computing FTLE. We demonstrate the efficiency of our method for double Gyre and ABC flows.