Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems

Abstract

In this paper we develop new techniques for revealing geometrical structures in phase space that are valid for aperiodically time dependent dynamical systems, which we refer to as Lagrangian descriptors. These quantities are based on the integration, for a finite time, along trajectories of an intrinsic bounded, positive geometrical and/or physical property of the trajectory itself. We discuss a general methodology for constructing Lagrangian descriptors, and we discuss a "heuristic argument" that explains why this method is successful for revealing geometrical structures in the phase space of a dynamical system. We support this argument by explicit calculations on a benchmark problem having a hyperbolic fixed point with stable and unstable manifolds that are known analytically. Several other benchmark examples are considered that allow us the assess the performance of Lagrangian descriptors in revealing invariant tori and regions of shear. Throughout the paper "side-by-side" comparisons of the performance of Lagrangian descriptors with both finite time Lyapunov exponents (FTLEs) and finite time averages of certain components of the vector field ('time averages") are carried out and discussed. In all cases Lagrangian descriptors are shown to be both more accurate and computationally efficient than these methods. We also perform computations for an explicitly three dimensional, aperiodically time-dependent vector field and an aperiodically time dependent vector field defined as a data set. Comparisons with FTLEs and time averages for these examples are also carried out, with similar conclusions as for the benchmark examples.

A dynamical systems perspective for a real-time response to a marine oil spill

“…[26,15]). The choice of this tool versus others mentioned above, such as for instance FTLE, is based on the advantages discussed in the literature [27,28,29,30]. These are related to its accuracy, ability to detect Lagrangian features of the true ocean state, computational 115 efficiency and programming simplicity.…”

“…The initial green circle approaches the coast line following the stable manifold, and elongates itself along the unstable manifold. Figure 2D) provides further details of the underlying skeleton, by plotting two unstable A skeleton of the stable and unstable manifolds of hyperbolic trajectories in time dependent flows can be constructed with the technique that is referred to as Lagrangian descriptors, based on the function M (see [40,26,27]), defined as follows: Another tool for revealing invariant manifolds in time dependent flows, which has been extensively applied in geophysical contexts [12,25,9] In order to deal with Eq. (1), we have assumed that particle motion is mainly horizontal on a sphere of radius R:…”

Accurate control of hyperbolic trajectories in any dimension

“…The importance of the large class of hyperbolic trajectories outlined above is underscored by their effect on the global behavior. For example, it is well known that stable and unstable manifolds form important flow separators in fluid flows, thereby providing a skeleton which distinguishes between regions of anomalous motion [4,[6][7][8][9][10]. Hyperbolic trajectories are entities to which these stable and unstable manifolds are attached, and hence their motion with time governs the time variation of these flow separators.…”

Variational principle for the determination of unstable periodic orbits and instanton trajectories at saddle points

“…We have found that such POs can be obtained through a variational procedure based on Lagrangian descriptors (LDs). [36][37][38][39][40][41][42][43][44] The connection of the LD to the construction of POs is presented in Sec. II.…”