Global phase space structures in a model of passive descent

TL;DR

This paper investigates the global phase space structures of a simplified model of passive descent, revealing an attracting manifold that organizes the trajectories and generalizes the concept of terminal velocity.

Contribution

It introduces the concept of a terminal velocity manifold in extended phase space and provides methods to approximate it and the associated dynamics.

Findings

Identification of a dominant attracting invariant manifold in velocity space.

Existence of an equilibrium submanifold with alternating stability.

Development of theoretical and numerical methods for manifold approximation.

Abstract

Even the most simplified models of falling and gliding bodies exhibit rich nonlinear dynamical behavior. Taking a global view of the dynamics of one such model, we find an attracting invariant manifold that acts as the dominant organizing feature of trajectories in velocity space. This attracting manifold captures the final, slowly changing phase of every passive descent, providing a higher-dimensional analogue to the concept of terminal velocity, the terminal velocity manifold. Within the terminal velocity manifold in extended phase space, there is an equilibrium submanifold with equilibria of alternating stability type, with different stability basins. In this work, we present theoretical and numerical methods for approximating the terminal velocity manifold and discuss ways to approximate falling and gliding motion in terms of these underlying phase space structures.