Lagrangian descriptors for two dimensional, area preserving, autonomous and nonautonomous maps

Abstract

In this paper we generalize the method of Lagrangian descriptors to two dimensional, area preserving, autonomous and nonautonomous discrete time dynamical systems. We consider four generic model problems-a hyperbolic saddle point for a linear, area-preserving autonomous map, a hyperbolic saddle point for a nonlinear, areapreserving autonomous map, a hyperbolic saddle point for linear, area-preserving nonautonomous map, and a hyperbolic saddle point for nonlinear, area-preserving nonautonomous map. The discrete time setting allows us to evaluate the expression for the Lagrangian descriptors explicitly for a certain class of norms. This enables us to provide a rigorous setting for the notion that the 'singular sets" of the Lagrangian descriptors correspond to the stable and unstable manifolds of hyperbolic invariant sets, as well as to understand how this depends upon the particular norms that are used. Finally we analyze, from the computational point of view, the performance of this tool for general nonlinear maps, by computing the "chaotic saddle" for autonomous and nonautonomous versions of the Hénon map.

Lagrangian descriptors for open maps

“…In order to throw light on this, we modify the original definition of discrete LDs for area preserving maps. If we consider a trajectory {q t , p t } t=T t=−T , where t ∈ N, discrete LDs were defined in [8] as…”

“…Also, LDs have been successfully implemented [6] within the so-called geometric transition state theory to the identification of recrossingfree dividing surfaces, thus helping in the computation of chemical reaction rates, and the reactive islands that account for nonstatistical behavior in chemical reactions [7]. Moreover, the concept has been adapted to discrete dynamical systems like bidimensional area preserving maps [8], under the name of discrete LDs. In this work the singular sets of LDs have been associated to the invariant manifolds of some prototypical maps and a chaotic saddle has been identified.…”

A Theoretical Framework for Lagrangian Descriptors

“…Recently [Lopesino et al, 2015] have provided rigorous proofs in the framework of discrete maps, where it is precisely defined what is meant by the phrase "singular features". One of the goals of this article is to extend those results to continuous time dynamical systems.…”

“…One of the goals of this article is to extend those results to continuous time dynamical systems. In order to simplify the demonstrations, this paper provides a new way of constructing Lagrangian descriptors in the same spirit as in [Lopesino et al, 2015]. The idea is based on considering the p-norm of each velocity component, instead of the p-norm of the modulus of the velocity.…”

A simple kinematic model for the Lagrangian description of relevant nonlinear processes in the stratospheric polar vortex

“…The capability of LDs, in general, and M in particular, to reveal invariant manifolds was analysed in detail in Mancho et al (2013). Lopesino et al (2015) and Lopesino et al (2017) have discussed, in discrete and continuous-time dynamical systems, respectively, a theoretical framework for some particular versions of LDs in specific examples.…”