arXiv subjects

Stephen Wiggins

Publications and source records attributed to Stephen Wiggins.

At least 55 records · Page 3Linked to original sources

arXiv2021

Transport and roaming on the Double van der Waals Potential Energy Surface

This paper explores the phase space structures characterising transport for a double-well van der Waals potential surface. Trajectories are classified as inter-well, intra-well, and escaping-from-a-well to define different dynamical fates. In particular, roaming trajectories, which are a new paradigm in chemical reaction dynamics, are observed. We apply the method of Lagrangian descriptors to uncover the boundaries between the fates. We find that the boundaries between different fates are the stable and unstable manifolds of the hyperbolic periodic orbits, which bifurcate when the energy increases. The methodology presented here applies to other open systems with a double-well potential.

nlin.CD↗

arXiv2021

A bridge between invariant dynamical structures and uncertainty quantification

We develop a new quantifier for forward time uncertainty for trajectories that are solutions of models generated from data sets. Our uncertainty quantifier is defined on the phase space in which the trajectories evolve and we show that it has a rich structure that is directly related to phase space structures from dynamical systems theory, such as hyperbolic trajectories and their stable and unstable manifolds. We apply our approach to an ocean data set, as well as standard benchmark models from deterministic dynamical systems theory. A significant application of our results, is that they allow a quantitative comparison of the transport performance described from different ocean data sets. This is particularly interesting nowadays when a wide variety of sources are available, since our methodology provides avenues for assessing the effective use of these data sets in a variety of situations.

math.DS↗

arXiv2021

Hamiltonian pitchfork bifurcation in transition across index-1 saddles

We study the effect of changes in the parameters of a two-dimensional potential energy surface on the phase space structures relevant for chemical reaction dynamics. The changes in the potential energy are representative of chemical reactions such as isomerization between two structural conformations or dissociation of a molecule with an intermediate. We present a two degrees of freedom quartic Hamiltonian that shows pitchfork bifurcation when the parameters are varied and we derive the bifurcation criteria relating the parameters. Next, we describe the phase space structures - unstable periodic orbits and their associated invariant manifolds, and phase space dividing surfaces - for the systems that can show trajectories undergo reaction defined as crossing of a potential energy barrier. Finally, we quantify the reaction dynamics for these systems by obtaining the directional flux and gap time distribution to illustrate the dependence on total energy and the coupling strength between the two degrees of freedom.

math.DS↗

arXiv2020

Phase space structure and escape time dynamics in a Van der Waals model for exothermic reactions

We study the phase space structures that control the transport in a classical Hamiltonian model for a chemical reaction. This model has been proposed to study the yield of products in an ultracold exothermic reaction. In the considered model, two elements determine the evolution of the system: a Van der Waals force and short-range force associated with the many-body interactions. In the previous work has been used small random periodic changes in the direction of the momentum to simulate the short-range many-body interactions. In the present work, random Gaussian bumps have been added to the Van der Waals potential energy simulate the short-range effects between the particles in the system. We compare both variants of the model and explain their differences similarities and differences from a phase space perspective. In order to visualize the structures that direct the dynamics in the phase space, we construct a natural Lagrangian descriptor for Hamiltonian systems based on the Maupertuis action $S_0$.

nlin.CD↗

arXiv2020

The tipping times in an Arctic sea ice system under influence of extreme events

In light of the rapid recent retreat of Arctic sea ice, the extreme weather events triggering the variability in Arctic ice cover has drawn increasing attention. A non-Gaussian $α$-stable Lévy process is thought to be an appropriate model to describe such extreme event. The maximal likely trajectory, based on the nonlocal Fokker-Planck equation, is applied to a nonautonomous Arctic sea ice system under $α$-stable Lévy noise. Two types of tipping times, the early-warning tipping time and the disaster-happening tipping time, are used to predict the critical time for the maximal likely transition from a perennially ice-covered state to a seasonally ice-free one, and from a seasonally ice-free state to a perennially ice-free one, respectively. We find that the increased intensity of extreme events results in shorter warning time for sea ice melting, and that an enhanced greenhouse effect will intensify this influence, making the arrival of warning time significantly earlier. Meanwhile, for the enhanced greenhouse effect, we discover that increased intensity and frequency of extreme events will advance the disaster-happening tipping time, in which an ice-free state is maintained throughout the year in the Arctic Ocean. Finally, we identify values of Lévy index $α$ and noise intensity $ε$ in $αε$-space that can trigger a transition between the Arctic sea ice state. These results provide an effective theoretical framework for studying Arctic sea ice variations under the influence of extreme events.

physics.ao-ph↗

arXiv2020

Elementary exposition of realizing phase space structures relevant to chemical reaction dynamics

In this article, we review the analytical and numerical approaches for computing the phase space structures in two degrees-of-freedom Hamiltonian systems that arise in chemical reactions. In particular, these phase space structures are the unstable periodic orbit associated with an index-1 saddle, the periodic orbit dividing surface, and the stable and unstable invariant manifolds of the unstable periodic orbit. We review the approaches in the context of a two degrees-of-freedom Hamiltonian with a quartic potential coupled with a quadratic potential. We derive the analytical form of the phase space structures for the integrable case and visualize their geometry on the three dimensional energy surface. We then investigate the bifurcation of the dividing surface due to the changes in the parameters of the potential energy. We also review the numerical method of \emph{turning point} and present its new modification called the \emph{turning point based on configuration difference} for computing the unstable periodic orbit in two degrees-of-freedom systems. These methods are implemented in the open-source python package, UPOsHam~\cite{Lyu2020}.

nlin.CD↗

arXiv2020

Using Lagrangian descriptors to uncover invariant structures in Chesnavich's Isokinetic Model with application to roaming

Complementary to existing applications of Lagrangian descriptors as an exploratory method, we use Lagrangian descriptors to find invariant manifolds in a system where some invariant structures have already been identified. In this case we use the parametrisation of a periodic orbit to construct a Lagrangian descriptor that will be locally minimised on its invariant manifolds. The procedure is applicable (but not limited) to systems with highly unstable periodic orbits, such as the isokinetic Chesnavich CH4+ model subject to a Hamiltonian isokinetic theromostat. Aside from its low computational requirements, the method enables us to study the invariant structures responsible for roaming in the isokinetic Chesnavich CH4+ model.

math.DS↗

arXiv2020

The role of depth and flatness of a potential energy surface in chemical reaction dynamics

In this study, we analyze how changes in the geometry of a potential energy surface in terms of depth and flatness can affect the reaction dynamics. We formulate depth and flatness in the context of one and two degree-of-freedom (DOF) Hamiltonian normal form for the saddle-node bifurcation and quantify their influence on chemical reaction dynamics. In a recent work, García-Garrido, Naik, and Wiggins illustrated how changing the well-depth of a potential energy surface (PES) can lead to a saddle-node bifurcation. They have shown how the geometry of cylindrical manifolds associated with the rank-1 saddle changes en route to the saddle-node bifurcation. Using the formulation presented here, we show how changes in the parameters of the potential energy control the depth and flatness and show their role in the quantitative measures of a chemical reaction. We quantify this role of the depth and flatness by calculating the ratio of the bottleneck-width and well-width, reaction probability (also known as transition fraction or population fraction), gap time (or first passage time) distribution, and directional flux through the dividing surface (DS) for small to high values of total energy. The results obtained for these quantitative measures are in agreement with the qualitative understanding of the reaction dynamics.

physics.chem-ph↗

arXiv2020

Revealing Roaming on the Double Morse Potential Energy Surface with Lagrangian Descriptors

In this paper, we analyse the phase space structure of the roaming dynamics in a two degree of freedom potential energy surface consisting of two identical planar Morse potentials separated by a distance. This potential energy surface was previously studied in [1], and it has two potential wells surrounded by an unbounded flat region containing no critical points. We study the phase space mechanism for the transference between the wells using the method of Lagrangian descriptors.

nlin.CD↗

arXiv2020

Exploring Isomerization Dynamics on a Potential Energy Surface with an Index-2 Saddle using Lagrangian Descriptors

In this paper we explore the phase space structures governing isomerization dynamics on a potential energy surface with four wells and an index-2 saddle. For this model, we analyze the influence that coupling both degrees of freedom of the system and breaking the symmetry of the problem have on the geometrical template of phase space structures that characterizes reaction. To achieve this goal we apply the method of Lagrangian descriptors, a technique with the capability of unveiling the key invariant manifolds that determine transport processes in nonlinear dynamical systems. This approach reveals with extraordinary detail the intricate geometry of the isomerization routes interconnecting the different potential wells, and provides us with valuable information to distinguish between initial conditions that undergo sequential and concerted isomerization.

nlin.CD↗

arXiv2019

Roaming at Constant Kinetic Energy: Chesnavich's Model and the Hamiltonian Isokinetic Thermostat

We consider the roaming mechanism for chemical reactions under the nonholonomic constraint of constant kinetic energy. Our study is carried out in the context of the Hamiltonian isokinetic thermostat applied to Chesnavich's model for an ion-molecule reaction. Through an analysis of phase space structures we show that imposing the nonholonomic constraint does not prevent the system from exhibiting roaming dynamics, and that the origin of the roaming mechanism turns out to be analogous to that found in the previously studied Hamiltonian case.

nlin.CD↗

arXiv2019

Tilting and Squeezing: Phase space geometry of Hamiltonian saddle-node bifurcation and its influence on chemical reaction dynamics

In this article we present the influence of a Hamiltonian saddle-node bifurcation on the high-dimensional phase space structures that mediate reaction dynamics. To achieve this goal, we identify the phase space invariant manifolds using Lagrangian descriptors, which is a trajectory-based diagnostic suitable for the construction of a complete ``phase space tomography'' by means of analyzing dynamics on low-dimensional slices. First, we build a Hamiltonian system with one degree-of-freedom (DoF) that models reaction, and study the effect of adding a parameter to the potential energy function that controls the depth of the well. Then, we extend this framework to a saddle-node bifurcation for a two DoF Hamiltonian, constructed by coupling a harmonic oscillator, i.e. a bath mode, to the other reactive DoF in the system. For this problem, we describe the phase space structures associated with the rank-1 saddle equilibrium point in the bottleneck region, which is a Normally Hyperbolic Invariant Manifold (NHIM) and its stable and unstable manifolds. Finally, we address the qualitative changes in the reaction dynamics of the Hamiltonian system due to changes in the well depth of the potential energy surface that gives rise to the saddle-node bifurcation.

nlin.CD↗

arXiv2019

Dynamics of the Morse Oscillator: Analytical Expressions for Trajectories, Action-Angle Variables, and Chaotic Dynamics

We consider the one degree-of-freedom Hamiltonian system defined by the Morse potential energy function (the "Morse oscillator"). We use the geometry of the level sets to construct explicit expressions for the trajectories as a function of time, their period for the bounded trajectories, and action-angle variables. We use these trajectories to prove sufficient conditions for chaotic dynamics, in the sense of Smale horseshoes, for the time-periodically perturbed Morse oscillator using a Melnikov type approach.

math.DS↗

arXiv2019

Finding NHIM in 2 and 3 degrees-of-freedom with Hénon-Heiles type potential

We present the capability of Lagrangian descriptors for revealing the high dimensional phase space structures that are of interest in nonlinear Hamiltonian systems with index-1 saddle. These phase space structures include normally hyperbolic invariant manifolds and their stable and unstable manifolds, and act as codimenision-1 barriers to phase space transport. The method is applied to classical two and three degrees-of-freedom Hamiltonian systems which have implications for myriad applications in physics and chemistry.

math.DS↗

arXiv2019

Finding NHIM: Identifying High Dimensional Phase Space Structures in Reaction Dynamics using Lagrangian Descriptors

Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, their stable and unstable manifolds have been an integral part of computing quantitative results such as transition fraction, stability erosion in multi-stable mechanical systems, and reaction rates in chemical reaction dynamics. Thus, methods that can reveal their geometry in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In this study, we assess the capability of one such method called Lagrangian descriptor for revealing the types of high dimensional phase space structures associated with index-1 saddle in Hamiltonian systems. The Lagrangian descriptor based approach is applied to two and three degree-of-freedom quadratic Hamiltonian systems where the high dimensional phase space structures are known, that is as closed-form analytical expressions. This leads to a direct comparison of features in the Lagrangian descriptor plots and the phase space structures' intersection with an isoenergetic two-dimensional surface and hence provides a validation of the approach.

math.DS↗

arXiv2018

Phase Space Analysis of the Non-Existence of Dynamical Matching in a Stretched Caldera Potential Energy Surface

In this paper we continue our studies of the two dimensional caldera potential energy surface in a parametrized family that allows for a study of the effect of symmetry on the phase space structures that govern how trajectories enter, cross, and exit the region of the caldera. As a particular form of trajectory crossing, we are able to determine the effect of symmetry and phase space structure on dynamical matching. We show that there is a critical value of the symmetry parameter which controls the phase space structures responsible for the manner of crossing, interacting with the central region (including trapping in this region) and exiting the caldera. We provide an explanation for the existence of this critical value in terms of the behavior of the Henon stability parameter for the associated periodic orbits.

nlin.CD↗

arXiv2018

Lagrangian study of the final warming in the southern stratosphere during 2002: Part II. 3D structure

This two-part paper aims to provide a Lagrangian perspective of the final southern warming in spring of 2002, during which the stratospheric polar vortex (SPV) experienced a unique splitting. We approach the subject from a dynamical systems viewpoint and search for Lagrangian coherent structures using a Lagrangian descriptor that is applied to reanalysis data. Part I presents our methodology and focuses by means of a kinematic model, on the understanding of fundamental processes for filamentation and ultimately for vortex splitting on an isentropic surface in the middle stratosphere. The present Part II discusses the three dimensional (3D) evolution of the flow during the selected event. For this, we apply concepts developed in Part I concerning a definition of the vortex boundary that helps in the selection of trajectories to illuminate the evolving flow structures, and a criterion that allows to justify why at an isentropic level a pinched vortex will split in later times. Lagrangian structures identified include surfaces that are several kilometers deep, and which a particle trajectory analysis confirms as barriers to the flow. The role of Lagrangian structures in determining the fate of particles during the SPV splitting is discussed.

physics.ao-ph↗

arXiv2018

Lagrangian study of the final warming in the southern stratosphere during 2002: Part I. The Vortex Splitting at Upper Levels

The present two-part paper provides a Lagrangian perspective of the final southern warming in 2002, during which the stratospheric polar vortex (SPV) experienced a unique splitting. Part I focuses on the understanding of fundamental processes for filamentation and ultimately for vortex splitting on a selected isentropic surface in the middle stratosphere. Part II discusses the three-dimensional evolution of the selected sudden warming event. We approach the subject from a dynamical systems viewpoint and search for Lagrangian coherent structures using a Lagrangian descriptor as a tool. In this Part I we work in the idealized framework of a kinematic model that allows for an understanding of the contributing elements of the flow in late September during the splitting. We introduce a definition of kinematic SPV boundary based on a criterion for binning parcels inside and outside the vortex according to their values of the Lagrangian descriptor and associated PDF. This definition is justified by using arguments that go beyond heuristic considerations based on the potential vorticity. Next, we turn to the filamentation proceses along this kinematic boundary, and the role of Lagrangian structures in the SPV splitting. We determine a criterion for splitting based on the structure of the evolving unstable and unstable manifolds.

physics.ao-ph↗