arXiv subjects

Shane D. Ross

Publications and source records attributed to Shane D. Ross.

25 records · Page 2Linked to original sources

arXiv2019

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

This paper introduces the trajectory divergence rate, a scalar field which locally gives the instantaneous attraction or repulsion of adjacent trajectories. This scalar field may be used to find highly attracting or repelling invariant manifolds, such as slow manifolds, to rapidly approximating hyperbolic Lagrangian coherent structures, or to provide the local stability of invariant manifolds. This work presents the derivation of the trajectory divergence rate and the related trajectory divergence ratio for 2-dimensional systems, investigates their properties, shows their application to several example systems, and presents their extension to higher dimensions.

nlin.CD↗

arXiv2019

A Tube Dynamics Perspective Governing Stability Transitions: An Example Based on Snap-through Buckling

The equilibrium configuration of an engineering structure, able to withstand a certain loading condition, is usually associated with a local minimum of the underlying potential energy. However, in the nonlinear context, there may be other equilibria present, and this brings with it the possibility of a transition to an alternative (remote) minimum. That is, given a sufficient disturbance, the structure might buckle, perhaps suddenly, to another shape. This paper considers the dynamic mechanisms under which such transitions (typically via saddle points) occur. A two-mode Hamiltonian is developed for a shallow arch/buckled beam. The resulting form of the potential energy---two stable wells connected by rank-1 saddle points---shows an analogy with resonance transitions in celestial mechanics or molecular reconfigurations in chemistry, whereas here the transition corresponds to switching between two stable structural configurations. Then, from Hamilton's equations, the analytical equilibria are determined and linearization of the equations of motion about the saddle is obtained. After computing the eigenvalues and eigenvectors of the coefficient matrix associated with the linearization, a symplectic transformation is given which puts the Hamiltonian into normal form and simplifies the equations, allowing us to use the conceptual framework known as tube dynamics. The flow in the equilibrium region of phase space as well as the invariant manifold tubes in position space are discussed. Also, we account for the addition of damping in the tube dynamics framework, which leads to a richer set of behaviors in transition dynamics than previously explored.

nlin.CD↗

arXiv2018

Experimental validation of phase space conduits of transition between potential wells

A phase space boundary between transition and non-transition, similar to those observed in chemical reaction dynamics, is shown experimentally in a macroscopic system. We present a validation of the phase space flux across rank one saddles connecting adjacent potential wells and confirm the underlying phase space conduits that mediate the transition. Experimental regions of transition are found to agree with the theory to within 1\%, suggesting the robustness of phase space conduits of transition in a broad array of two or more degree of freedom experimental systems, despite the presence of small dissipation.

nlin.CD↗

arXiv2017

Computational Method for Phase Space Transport with Applications to Lobe Dynamics and Rate of Escape

Lobe dynamics and escape from a potential well are general frameworks introduced to study phase space transport in chaotic dynamical systems. While the former approach studies how regions of phase space are transported by reducing the flow to a two-dimensional map, the latter approach studies the phase space structures that lead to critical events by crossing periodic orbit around saddles. Both of these frameworks require computation with curves represented by millions of points-computing intersection points between these curves and area bounded by the segments of these curves-for quantifying the transport and escape rate. We present a theory for computing these intersection points and the area bounded between the segments of these curves based on a classification of the intersection points using equivalence class. We also present an alternate theory for curves with nontransverse intersections and a method to increase the density of points on the curves for locating the intersection points accurately.The numerical implementation of the theory presented herein is available as an open source software called Lober. We used this package to demonstrate the application of the theory to lobe dynamics that arises in fluid mechanics, and rate of escape from a potential well that arises in ship dynamics.

math.DS↗

arXiv2013

Demonstration of Experimental Three Dimensional Finite Time Lyapunov Exponents with Inertial Particles

This work provides an experimental method for simultaneously measuring finite time Lyapunov exponent fields for multiple particle groups, including non-flow tracers, in three-dimensional multiphase flows. From sequences of particle images, e.g., from experimental fluid imaging techniques, we can directly compute the flow map and coherent structures, with out performing the computationally costly numerical integration. This is particularly useful to find three-dimensional Lagrangian coherent structures for inertial particles, that do not follow the bulk fluid velocity, as we demonstrate for a grid turbulence experiment. The technique described may provide a new means for exploring the physics of experimental multi-phase flows.

physics.flu-dyn↗

arXiv2012

Topological chaos, braiding and bifurcation of almost-cyclic sets

In certain two-dimensional time-dependent flows, the braiding of periodic orbits provides a way to analyze chaos in the system through application of the Thurston-Nielsen classification theorem (TNCT). We expand upon earlier work that introduced the application of the TNCT to braiding of almost-cyclic sets, which are individual components of almost-invariant sets [Stremler, Ross, Grover, Kumar, Topological chaos and periodic braiding of almost-cyclic sets. Physical Review Letters 106 (2011), 114101]. In this context, almost-cyclic sets are periodic regions in the flow with high local residence time that act as stirrers or `ghost rods' around which the surrounding fluid appears to be stretched and folded. In the present work, we discuss the bifurcation of the almost-cyclic sets as a system parameter is varied, which results in a sequence of topologically distinct braids. We show that, for Stokes' flow in a lid-driven cavity, these various braids give good lower bounds on the topological entropy over the respective parameter regimes in which they exist. We make the case that a topological analysis based on spatiotemporal braiding of almost-cyclic sets can be used for analyzing chaos in fluid flows. Hence we further develop a connection between set-oriented statistical methods and topological methods, which promises to be an important analysis tool in the study of complex systems.

nlin.CD↗

arXiv2007

Separatrices and basins of stability from time series data

An approach is presented for identifying separatrices in phase space generated from noisy time series data sets representative of measured experimental data. These separatrices are identified as ridges in the phase space distribution of finite-time Lyapunov exponents, i.e., Lagrangian coherent structures (LCS). As opposed to previous approaches, the LCS is identified using only trajectories since no analytical or data-defined vector field is available. The method is applied to a biological simulation in which the separatrix reveals a basin of stability. These results suggest that the method will be a fruitful approach to time series analysis, particularly in cases where a limited number of trajectories are available as might be encountered in experiments.

nlin.CD↗