Global phase space structures in a model of passive descent

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.in the work of Maxwell and Zhukovskii [12][13][14]. Prior experimental studies have identified several canonical behaviors exhibited by falling disks, plates, and plant seeds characterized by different couplings between rotation and translation in space [2,15]. In all of these behaviors, an inherently high-dimensional system, which must consider velocities, angles, and angular velocities in 3D space, converges to a low-dimensional behavior, whether traveling in a single plane or going through a cycle of velocities [15][16][17][18][19][20].Mathematical models have offered many insights into passive descent. Ideal flow theory has been used to study the motion of a body both through a steady fluid and interacting with shed vortices [13,[21][22][23]. The Zhukovskii problem or phugoid model, which assumes that the wing travels with constant angle-of-attack, is a two-dimensional ordinary differential equation for flight from which phugoid oscillations, which couple forward velocity with pitch angle, arise [14]. Andersen et al. developed a phenomenological model based on experiments and simulations which produce fluttering, tumbling, and even chaotic behavior through a 4D differential equation [17,18]. To focus specifically on gliding animal flight and compare the gliding capabilities of different animals, Yeaton et al. [24] introduced a two-dimensional model for non-equilibrium gliding of animals. It is a modification of this model which we will consider in the present work. In this model, the authors decoupled translational and rotational dynamics in order to take a deeper view of the translational behavior and shape dependence based on lift and drag characteristics alone. To do so, inspired by the motion of gliding animals, the authors treat pitch angle as fixed with respect to the ground, assuming that the glider has some amount of control to hold this angle. Lift and drag are treated as functional parameters in this model, which is used to capture the differences in glider shapes. This model works especially well for gliding animals, but may be extended to general passive descent. Yeaton et al. [24] analyzed a variety of animal gliders and found tha...

Trajectory-free approximation of phase space structures using the trajectory divergence rate

“…In a simplified model of passive gliding flight, a globally attracting codimension-one manifold may be observed in the glider's velocity space [30,47]. Because every trajectory is rapidly attracted to this structure and evolves along or near it, it serves as a higher dimensional analogue to terminal velocity and is therefore referred to as the terminal velocity manifold.…”

“…From the stable fixed point located at (v x , v z ) = (0.50, −0.56), there are both strong and weak stable submanifolds within the 2dimensional stable manifold. Because many invariant manifolds intersect the origin with the same tangent direction, the weak stable submanifold is nonunique, and methods such as the trajectory-normal repulsion rate may be used to identify the most influential weak stable submanifold [20,30].…”

“…There often exist lower-dimensional manifolds which dominate the attraction and repulsion, swirling, or shearing of trajectories advecting under the flow. Methods to find such structures have been applied to better understand topics including plant pathogen spread [34], animal locomotion [30,32], seabird foraging patterns [22], geophysical flows [26,43], chemical reactions [48], comet distributions [11], and structural mechanics [45].…”

“…Stable, unstable, and center manifolds of a fixed point may be calculated through a number of classical methods, including "growing" the stable or unstable manifolds by integrating the eigendirections of fixed points backward or forward in time [23]. However, in the context of weak stable submanifolds, these methods begin to break down [30]. Some weak submanifolds are part of a class of geometric structures known as slow manifolds which exhibit a separation of time scales [25], attracting or repelling other trajectories in phase space.…”

“…Hyperbolic coherent structures represent dynamically evolving transport barriers in flows [36,37]. Although the methods of coherent structures are typically situated within the context of fluid dynamics, such structures have applications to the flow of general vector fields [1,15,30,40]. Detecting and analyzing the underlying structures of flows gives a better understanding of how the system evolves, whether that flow represents the motion of a fluid or some other general dynamical system.…”