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2013
Impact Factor

Stephen Wiggins

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

Publications:

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

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