Interpolating vector fields for near identity maps and averaging

TL;DR

This paper introduces a method to interpolate near identity maps with explicit vector fields using Lagrangian interpolation, enabling analysis of dynamics and adiabatic invariants, with applications to symplectic maps and Arnold diffusion.

Contribution

It develops a novel approach to construct interpolating vector fields for near identity maps, providing new tools for studying their dynamics and invariants.

Findings

Explicit interpolating vector fields for near identity maps

Construction of adiabatic invariants in symplectic maps

Visualization of high-dimensional map dynamics

Abstract

For a smooth near identity map, we introduce the notion of an interpolating vector field written in terms of iterates of the map. Our construction is based on Lagrangian interpolation and provides an explicit expressions for autonomous vector fields which approximately interpolate the map. We study properties of the interpolating vector fields and explore their applications to the study of dynamics. In particular, we construct adiabatic invariants for symplectic near identity maps. We also introduce the notion of a Poincar\'e section for a near identity map and use it to visualise dynamics of four dimensional maps. We illustrate our theory with several examples, including the Chirikov standard map and a symplectic map in dimension four, an example motivated by the theory of Arnold diffusion.