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, area-preserving 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.

References in corpus (6)

• Lagrangian Descriptors: A Method for Revealing Phase Space Structures of General Time Dependent Dynamical Systems
• Distinguished trajectories in time dependent vector fields
• The hidden geometry of ocean flows
• The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current
• The turnstile mechanism across the Kuroshio current: analysis of dynamics in altimeter velocity fields
• Coherent structures and the saturation of a nonlinear dynamo

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