Identifying invariant manifold using phase space warping and stochastic interrogation

Advances in the generalization of invariant manifolds to finite time, experimental (or observational) flows have stimulated many recent developments in the approximation of invariant manifolds and Lagrangian coherent structures. This paper explores the identification of invariant manifold like structures in experimental settings, where knowledge of a flow field is absent, but phase space trajectories can be experimentally measured. Several existing methods for the approximation of these structures modified for application when only unstructured trajectory data is available. We find the recently proposed method, based on the concept of phase space warping, to outperform other methods as data becomes limited and show it to extend the finite-time Lyapunov exponent method. This finding is based on a comparison of methods for various data quantities and in the presence of both measurement and dynamic noise.