Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems

TL;DR

This paper introduces Lagrangian descriptors, a new method for revealing phase space structures in time-dependent dynamical systems, demonstrating their effectiveness and efficiency over existing techniques through various benchmark and real-world examples.

Contribution

The paper develops a novel technique called Lagrangian descriptors for analyzing phase space structures in aperiodic dynamical systems, outperforming traditional methods in accuracy and computational efficiency.

Findings

Lagrangian descriptors accurately reveal invariant manifolds and tori.

They outperform FTLEs and time averages in accuracy and efficiency.

Effective in both 2D and 3D aperiodic 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.