Measure, dimension, and complexity of the transient motion in Hamiltonian systems

TL;DR

This paper introduces a numerical method to visualize and quantify transient motion in Hamiltonian systems, revealing how initial conditions influence escape dynamics through geometrical and dynamical measures.

Contribution

The work presents a novel numerical approach to analyze transient behavior in Hamiltonian systems using the transient measure, applied to tokamak and celestial models.

Findings

Transient measures effectively characterize escape dynamics.

Different initial conditions lead to diverse transient scenarios.

Transient correlation dimension and complexity coefficient quantify these scenarios.

Abstract

Hamiltonian systems that are either open, leaking, or contain holes in the phase space possess solutions that eventually escape the system's domain. The motion described by such escape orbits before crossing the escape threshold can be understood as a transient behavior. In this work, we introduce a numerical method to visually illustrate and quantify the transient motion in Hamiltonian systems based on the transient measure, a finite-time version of the natural measure. We apply this method to two physical systems: the single-null divertor tokamak, described by a symplectic map; and the Earth-Moon system, as modeled by the planar circular restricted three-body problem. Our results portray how different locations for the ensemble of initial conditions may lead to different transient dynamical scenarios in both systems. We show that these scenarios can be properly quantified from a geometrical aspect, the transient correlation dimension, and a dynamical aspect, the transient complexity coefficient.