Global phase-space geometry of three-dimensional gliding: terminal velocity manifolds, separatrices, and stability structure

Abstract

We develop a three-dimensional dynamical-systems framework for passive gliding and identify the global phase-space structures that organize its motion. Extending previous two-dimensional models of non-equilibrium gliding, we show that the 3D velocity dynamics possess an attracting, normally hyperbolic invariant surface, the terminal velocity manifold (TVM), onto which all trajectories rapidly collapse before evolving slowly toward a glide equilibrium. There is also a separatrix surface associated with an invariant manifold of an unstable equilibrium within the TVM, which partitions initial conditions into qualitatively distinct descent behaviors: efficient shallow glides versus steep, drag-dominated descent. Using lift-drag data from three representative airfoils--a snake-inspired bluff body, the Zimmerman planform characteristic of Draco lizards, and the classical NACA 0012--we compute the full equilibrium surfaces, analyze their pitch-roll bifurcations, and reconstruct the TVM and separatrix geometry in three dimensions. The results reveal that (i) equilibrium stability changes with both pitch and roll, rather than pitch alone; (ii) separatrix geometry determines the dynamic accessibility of shallow glides; and (iii) bio-inspired airfoils possess compact separatrix regions that make efficient gliding robust across a wide range of initial jump conditions. This work unifies biological and engineered gliders within a single global geometric framework and establishes separatrix geometry on the TVM as a principled diagnostic for glide robustness.