Stephen Wiggins

Senate House, Tyndall Avenue, Bristol BS8 1TH, UK
University of Bristol


Agaoglou M., Katsanikas M., Wiggins S.
In this paper we study an asymmetric valley-ridge inflection point (VRI) potential, whose energy surface (PES) features two sequential index-1 saddles (the upper and the lower), with one saddle having higher energy than the other, and two potential wells separated by the lower index-1 saddle. We show how the depth and the flatness of our potential changes as we modify the parameter that controls the asymmetry as well as how the branching ratio (ratio of the trajectories that enter each well) is changing as we modify the same parameter and its correlation with the area of the lobes as they have been formed by the stable and unstable manifolds that have been extracted from the gradient of the LD scalar fields.
Keywords: phase space structure, Lagrangian descriptors, chemical reaction dynamics, valley ridge inflection point potential
Citation: Agaoglou M., Katsanikas M., Wiggins S.,  The Influence of a Parameter that Controls the Asymmetry of a Potential Energy Surface with an Entrance Channel and Two Potential Wells, Regular and Chaotic Dynamics, 2022, vol. 27, no. 2, pp. 232-241
Haigh D., Katsanikas M., Agaoglou M., Wiggins S.
Selectivity is an important phenomenon in chemical reaction dynamics. This can be quantified by the branching ratio of the trajectories that visit one or the other well to the total number of trajectories in a system with a potential with two sequential index-1 saddles and two wells (top well and bottom well). In our case, the relative branching ratio is 1:1 because of the symmetry of our potential energy surface. The mechanisms of transport and the behavior of the trajectories in this kind of systems have been studied recently. In this paper we study the time evolution after the selectivity as energy varies using periodic orbit dividing surfaces. We investigate what happens after the first visit of a trajectory to the region of the top or the bottom well for different values of energy. We answer the natural question: What is the destiny of these trajectories?
Keywords: phase space structure, dividing surfaces, chemical physics, periodic orbits, homoclinic and heteroclinic orbits
Citation: Haigh D., Katsanikas M., Agaoglou M., Wiggins S.,  The Time Evolution of the Trajectories After the Selectivity in a Symmetric Potential Energy Surface with a Post-transition-state Bifurcation, Regular and Chaotic Dynamics, 2021, vol. 26, no. 6, pp. 763-774
Crossley R., Agaoglou M., Katsanikas M., Wiggins S.
In this paper we compare the method of Lagrangian descriptors with the classical method of Poincaré maps for revealing the phase space structure of two-degree-of-freedom Hamiltonian systems. The comparison is carried out by considering the dynamics of a twodegree- of-freedom system having a valley ridge inflection point (VRI) potential energy surface. VRI potential energy surfaces have four critical points: a high energy saddle and a lower energy saddle separating two wells. In between the two saddle points is a valley ridge inflection point that is the point where the potential energy surface geometry changes from a valley to a ridge. The region between the two saddles forms a reaction channel and the dynamical issue of interest is how trajectories cross the high energy saddle, evolve towards the lower energy saddle, and select a particular well to enter. Lagrangian descriptors and Poincaré maps are compared for their ability to determine the phase space structures that govern this dynamical process.
Keywords: phase space structure, periodic orbits, stable and unstable manifolds, homoclinic and heteroclinic orbits, Poincar´e maps, Lagrangian descriptors
Citation: Crossley R., Agaoglou M., Katsanikas M., Wiggins S.,  From Poincaré Maps to Lagrangian Descriptors: The Case of the Valley Ridge Inflection Point Potential, Regular and Chaotic Dynamics, 2021, vol. 26, no. 2, pp. 147-164
Lyu W., Naik S., Wiggins S.
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 saddlenode bifurcation and quantify their influence on chemical reaction dynamics [1, 2]. In a recent work, García-Garrido et al. [2] 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.
Keywords: Hamiltonian dynamics, bifurcation theory, phase space methods
Citation: Lyu W., Naik S., Wiggins S.,  The Role of Depth and Flatness of a Potential Energy Surface in Chemical Reaction Dynamics, Regular and Chaotic Dynamics, 2020, vol. 25, no. 5, pp. 453-475
Krajňák V., Ezra G. S., Wiggins S.
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.
Keywords: nonholonomic constraint, phase space structures, invariant manifolds, chemical reaction, roaming
Citation: Krajňák V., Ezra G. S., Wiggins S.,  Roaming at Constant Kinetic Energy: Chesnavich's Model and the Hamiltonian Isokinetic Thermostat, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 615-627
García Garrido V. J., Balibrea-Iniesta F., Wiggins S., Mancho A. M., Lopesino C.
The goal of this paper is to apply Lagrangian Descriptors (LDs), a technique based on Dynamical Systems Theory (DST) to reveal the phase space structures present in the wellknown Arnold’s cat map. This discrete dynamical system, which represents a classical example of an Anosov diffeomorphism that is strongly mixing, will provide us with a benchmark model to test the performance of LDs and their capability to detect fixed points, periodic orbits and their stable and unstable manifolds present in chaotic maps. In this work we show, both from a theoretical and a numerical perspective, how LDs reveal the invariant manifolds of the periodic orbits of the cat map. The application of this methodology in this setting clearly illustrates the chaotic behavior of the cat map and highlights some technical numerical difficulties that arise in the identification of its phase space structures.
Keywords: dynamical systems, maps, Lagrangian descriptors, chaotic sets, stable and unstable manifolds, mixing
Citation: García Garrido V. J., Balibrea-Iniesta F., Wiggins S., Mancho A. M., Lopesino C.,  Detection of Phase Space Structures of the Cat Map with Lagrangian Descriptors, Regular and Chaotic Dynamics, 2018, vol. 23, no. 6, pp. 751-766
García Garrido V. J., Curbelo J., Mancho A. M., Wiggins S., Mechoso C. R.
The Application of Lagrangian Descriptors to 3D Vector Fields
2018, vol. 23, no. 5, pp.  551-568
Since the 1980s, the application of concepts and ideas from dynamical systems theory to analyze phase space structures has provided a fundamental framework to understand long-term evolution of trajectories in many physical systems. In this context, for the study of fluid transport and mixing the development of Lagrangian techniques that can capture the complex and rich dynamics of time-dependent flows has been crucial. Many of these applications have been to atmospheric and oceanic flows in two-dimensional (2D) relevant scenarios. However, the geometrical structures that constitute the phase space structures in time-dependent three-dimensional (3D) flows require further exploration. In this paper we explore the capability of Lagrangian descriptors (LDs), a tool that has been successfully applied to time-dependent 2D vector fields, to reveal phase space geometrical structures in 3D vector fields. In particular, we show how LDs can be used to reveal phase space structures that govern and mediate phase space transport. We especially highlight the identification of normally hyperbolic invariant manifolds (NHIMs) and tori. We do this by applying this methodology to three specific dynamical systems: a 3D extension of the classical linear saddle system, a 3D extension of the classical Duffing system, and a geophysical fluid dynamics f-plane approximation model which is described by analytical wave solutions of the 3D Euler equations. We show that LDs successfully identify and recover the template of invariant manifolds that define the dynamics in phase space for these examples.
Keywords: Lagrangian descriptors, phase space structure, invariant manifolds, invariant tori, ergodic decomposition
Citation: García Garrido V. J., Curbelo J., Mancho A. M., Wiggins S., Mechoso C. R.,  The Application of Lagrangian Descriptors to 3D Vector Fields, Regular and Chaotic Dynamics, 2018, vol. 23, no. 5, pp. 551-568
Carpenter B. K., Ezra G. S., Farantos S. C., Kramer Z. C., Wiggins S.
In this paper we analyze a two-degree-of-freedom Hamiltonian system constructed from two planar Morse potentials. The resulting potential energy surface has two potential wells surrounded by an unbounded flat region containing no critical points. In addition, the model has an index one saddle between the potential wells. We study the dynamical mechanisms underlying transport between the two potential wells, with emphasis on the role of the flat region surrounding the wells. The model allows us to probe many of the features of the “roaming mechanism” whose reaction dynamics are of current interest in the chemistry community.
Keywords: Double Morse potential, phase space structure, dynamics, periodic orbit, roaming
Citation: Carpenter B. K., Ezra G. S., Farantos S. C., Kramer Z. C., Wiggins S.,  Dynamics on the Double Morse Potential: A Paradigm for Roaming Reactions with no Saddle Points, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 60-79
Wiggins S.
In this paper we give an introduction to the notion of a normally hyperbolic invariant manifold (NHIM) and its role in chemical reaction dynamics. We do this by considering simple examples for one-, two-, and three-degree-of-freedom systems where explicit calculations can be carried out for all of the relevant geometrical structures and their properties can be explicitly understood. We specifically emphasize the notion of a NHIM as a “phase space concept”. In particular, we make the observation that the (phase space) NHIM plays the role of “carrying” the (configuration space) properties of a saddle point of the potential energy surface into phase space.
We also consider an explicit example of a 2-degree-of-freedom system where a “global” dividing surface can be constructed using two index one saddles and one index two saddle. Such a dividing surface has arisen in several recent applications and, therefore, such a construction may be of wider interest.
Keywords: normally hyperbolic invariant manifolds, chemical reaction dynamics, dividing surface, phase space transport, index $k$ saddle points
Citation: Wiggins S.,  The Role of Normally Hyperbolic Invariant Manifolds (NHIMs) in the Context of the Phase Space Setting for Chemical Reaction Dynamics, Regular and Chaotic Dynamics, 2016, vol. 21, no. 6, pp. 621-638
Fortunati A., Wiggins S.
The aim of this paper is to prove the Kolmogorov theorem of persistence of Diophantine flows for nearly integrable Poisson systems associated to a real analytic Hamiltonian with aperiodic time dependence, provided that the perturbation is asymptotically vanishing. The paper is an extension of an analogous result by the same authors for canonical Hamiltonian systems; the flexibility of the Lie series method developed by A. Giorgilli et al. is profitably used in the present generalization.
Keywords: Poisson systems, Kolmogorov theorem, aperiodic time dependence
Citation: Fortunati A., Wiggins S.,  A Kolmogorov Theorem for Nearly Integrable Poisson Systems with Asymptotically Decaying Time-dependent Perturbation, Regular and Chaotic Dynamics, 2015, vol. 20, no. 4, pp. 476-485
Cresson J., Wiggins S.
Let $N$ be a smooth manifold and $f:N \to N$ be a $C^l$, $l \geqslant 2$ diffeomorphism. Let $M$ be a normally hyperbolic invariant manifold, not necessarily compact. We prove an analogue of the $\lambda$-lemma in this case. Applications of this result are given in the context of normally hyperbolic invariant annuli or cylinders which are the basic pieces of all geometric mechanisms for diffusion in Hamiltonian systems. Moreover, we construct an explicit class of three-degree-of-freedom near-integrable Hamiltonian systems which satisfy our assumptions.
Keywords: $\lambda$-lemma, Arnold diffusion, normally hyperbolic manifolds, Moeckel's mechanism
Citation: Cresson J., Wiggins S.,  A $\lambda$-lemma for Normally Hyperbolic Invariant Manifolds, Regular and Chaotic Dynamics, 2015, vol. 20, no. 1, pp. 94-108
Fortunati A., Wiggins S.
The aim of this paper is to prove a Kolmogorov type result for a nearly integrable Hamiltonian, quadratic in the actions, with an aperiodic time dependence. The existence of a torus with a prefixed Diophantine frequency is shown in the forced system, provided that the perturbation is real-analytic and (exponentially) decaying with time. The advantage consists in the possibility to choose an arbitrarily small decaying coefficient consistently with the perturbation size.
The proof, based on the Lie series formalism, is a generalization of a work by A. Giorgilli.
Keywords: Hamiltonian systems, Kolmogorov theorem, aperiodic time dependence
Citation: Fortunati A., Wiggins S.,  Persistence of Diophantine Flows for Quadratic Nearly Integrable Hamiltonians under Slowly Decaying Aperiodic Time Dependence, Regular and Chaotic Dynamics, 2014, vol. 19, no. 5, pp. 586-600
Fortunati A., Wiggins S.
The aim of this paper is to extend the result of Giorgilli and Zehnder for aperiodic time dependent systems to a case of nearly integrable convex analytic Hamiltonians. The existence of a normal form and then a stability result are shown in the case of a slow aperiodic time dependence that, under some smallness conditions, is independent of the size of the perturbation.
Keywords: Hamiltonian systems, Nekhoroshev theorem, aperiodic time dependence
Citation: Fortunati A., Wiggins S.,  Normal Form and Nekhoroshev Stability for Nearly Integrable Hamiltonian Systems with Unconditionally Slow Aperiodic Time Dependence, Regular and Chaotic Dynamics, 2014, vol. 19, no. 3, pp. 363-373
Waalkens H., Wiggins S.
Hamiltonian dynamical systems possessing equilibria of saddle x center x∙∙∙x center stability type display reaction-type dynamics for energies close to the energy of such equilibria; entrance and exit from certain regions of the phase space is only possible via narrow bottlenecks created by the influence of the equilibrium points. In this paper we provide a thorough pedagogical description of the phase space structures that are responsible for controlling transport in these problems. Of central importance is the existence of a Normally Hyperbolic Invariant Manifold (NHIM), whose stable and unstable manifolds have sufficient dimensionality to act as separatrices, partitioning energy surfaces into regions of qualitatively distinct behavior. This NHIM forms the natural (dynamical) equator of a (spherical) dividing surface which locally divides an energy surface into two components ("reactants" and "products"), one on either side of the bottleneck. This dividing surface has all the desired properties sought for in transition state theory where reaction rates are computed from the flux through a dividing surface. In fact, the dividing surface that we construct is crossed exactly once by reactive trajectories, and not crossed by nonreactive trajectories, and related to these properties, minimizes the flux upon variation of the dividing surface.
We discuss three presentations of the energy surface and the phase space structures contained in it for 2-degree-of-freedom (DoF) systems in the three-dimensional space $\mathbb{R}^3$, and two schematic models which capture many of the essential features of the dynamics for $n$-DoF systems. In addition, we elucidate the structure of the NHIM.
Keywords: high dimensional Hamiltonian dynamics, phase space structure and geometry, normally hyperbolic invariant manifold, Poincaré–Birkhoff normal form theory, chemical reaction dynamics, transition state theory
Citation: Waalkens H., Wiggins S.,  Geometrical models of the phase space structures governing reaction dynamics, Regular and Chaotic Dynamics, 2010, vol. 15, no. 1, pp. 1-39
Rudnev M., Wiggins S.
On a Homoclinic Splitting Problem
2000, vol. 5, no. 2, pp.  227-242
We study perturbations of Hamiltonian systems of $n+1$ degrees of freedom $(n \geqslant 2)$ in the real-analytic case, such that in the absence of the perturbation they contain a partially hyperbolic (whiskered) $n$-torus with the Kronecker flow on it with a Diophantine frequency, connected to itself by a homoclinic exact Lagrangian submanifold (separatrix), formed by the coinciding unstable and stable manifolds (whiskers) of the torus. Typically, a perturbation causes the separatrix to split. We study this phenomenon as an application of the version of the KAM theorem, proved in [13]. The theorem yields the representations of global perturbed separatrices as exact Lagrangian submanifolds in the phase space. This approach naturally leads to a geometrically meaningful definition of the splitting distance, as the gradient of a scalar function on a subset of the configuration space, which satisfies a first order linear homogeneous PDE. Once this fact has been established, we adopt a simple analytic argument, developed in [15] in order to put the corresponding vector field into a normal form, convenient for further analysis of the splitting distance. As a consequence, we argue that in the systems, which are Normal forms near simple resonances for the perturbations of integrable systems in the action-angle variables, the splitting is exponentially small.
Citation: Rudnev M., Wiggins S.,  On a Homoclinic Splitting Problem, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 227-242
Rudnev M., Wiggins S.
On a Partially Hyperbolic KAM Theorem
1999, vol. 4, no. 4, pp.  39-58
We prove structural stability under small perturbations of a family of real analytic Hamiltonian systems of $n+1$ degrees of freedom $(n \geqslant 2)$, comprising an invariant partially hyperbolic $n$-torus with the Kronecker flow on it with a diophantine frequency, and an unstable (stable) exact Lagrangian submanifold (whisker), containing this torus. This is the preservation of the exact Lagrangian properties of the whisker that we focus upon. Hence, we develop a Normal form, which is valid globally in the neighborhood of the perturbed whisker and enables its representation as an exact Lagrangian submanifold in the original coordinates, whose generating function solves the Hamilton–Jacobi equation.
Citation: Rudnev M., Wiggins S.,  On a Partially Hyperbolic KAM Theorem, Regular and Chaotic Dynamics, 1999, vol. 4, no. 4, pp. 39-58

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