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2013
Impact Factor

# Stephen Wiggins

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

## Publications:

 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 2018, vol. 23, no. 1, pp.  60-79 Abstract 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 DOI:10.1134/S1560354718010069
 Wiggins S. The Role of Normally Hyperbolic Invariant Manifolds (NHIMs) in the Context of the Phase Space Setting for Chemical Reaction Dynamics 2016, vol. 21, no. 6, pp.  621-638 Abstract 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 DOI:10.1134/S1560354716060034
 Fortunati A., Wiggins S. A Kolmogorov Theorem for Nearly Integrable Poisson Systems with Asymptotically Decaying Time-dependent Perturbation 2015, vol. 20, no. 4, pp.  476-485 Abstract 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 DOI:10.1134/S1560354715040061
 Cresson J., Wiggins S. A $\lambda$-lemma for Normally Hyperbolic Invariant Manifolds 2015, vol. 20, no. 1, pp.  94-108 Abstract 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 DOI:10.1134/S1560354715010074
 Fortunati A., Wiggins S. Persistence of Diophantine Flows for Quadratic Nearly Integrable Hamiltonians under Slowly Decaying Aperiodic Time Dependence 2014, vol. 19, no. 5, pp.  586-600 Abstract 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 DOI:10.1134/S1560354714050062
 Fortunati A., Wiggins S. Normal Form and Nekhoroshev Stability for Nearly Integrable Hamiltonian Systems with Unconditionally Slow Aperiodic Time Dependence 2014, vol. 19, no. 3, pp.  363-373 Abstract 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 DOI:10.1134/S1560354714030071
 Waalkens H., Wiggins S. Geometrical models of the phase space structures governing reaction dynamics 2010, vol. 15, no. 1, pp.  1-39 Abstract 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 DOI:10.1134/S1560354710010016
 Rudnev M., Wiggins S. On a Homoclinic Splitting Problem 2000, vol. 5, no. 2, pp.  227-242 Abstract 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 DOI:10.1070/RD2000v005n02ABEH000146
 Rudnev M., Wiggins S. On a Partially Hyperbolic KAM Theorem 1999, vol. 4, no. 4, pp.  39-58 Abstract 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 DOI:10.1070/RD1999v004n04ABEH000130