Indicator Functions Detect Tangentially Transient Behaviour on Decaying Normally Hyperbolic Invariant Manifolds

TL;DR

This paper investigates how normally hyperbolic invariant manifolds (NHIMs) decay under perturbations in Hamiltonian systems, using indicator functions to detect tangential transient behaviors, exemplified by electron motion in a magnetic field.

Contribution

It introduces indicator functions to detect tangential transient effects during NHIM decay, linking decay dynamics with observable delay time indicators in Hamiltonian systems.

Findings

Decay of NHIMs can be tracked via indicator functions.

Tangential transient effects influence delay time indicators.

Application demonstrated on electron motion in magnetic dipole fields.

Abstract

We study the decay scenario of a codimension-2 NHIM in a three degrees of freedom Hamiltonian system under increasing perturbation when the NHIM loses its normal hyperbolicity. On one hand, we follow this decay in the Poincar\'e map for the internal dynamics of the NHIM. On the other hand, we also follow the decay in a time delay function calculated on a 2-dimensional plane in the phase space of the system. In addition, we observe the role of tangential transient effects on the decaying NHIM and their manifestation in the delay time indicator function. Thereby we obtain ideas on how the decay of NHIMs and the tangential transient effects are encoded in indicator functions. As an example of demonstration, we use the motion of an electron in a perturbed magnetic dipole field.