Stephen Wiggins
Publications:
García Garrido V. J., BalibreaIniesta F., Wiggins S., Mancho A. M., Lopesino C.
Detection of Phase Space Structures of the Cat Map with Lagrangian Descriptors
2018, vol. 23, no. 6, pp. 751766
Abstract
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.

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. 551568
Abstract
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
longterm 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 timedependent flows has been crucial. Many of these
applications have been to atmospheric and oceanic flows in twodimensional (2D) relevant
scenarios. However, the geometrical structures that constitute the phase space structures in
timedependent threedimensional (3D) flows require further exploration. In this paper we
explore the capability of Lagrangian descriptors (LDs), a tool that has been successfully
applied to timedependent 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 fplane
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.

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. 6079
Abstract
In this paper we analyze a twodegreeoffreedom 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.

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. 621638
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 threedegreeoffreedom 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 2degreeoffreedom 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. 
Fortunati A., Wiggins S.
A Kolmogorov Theorem for Nearly Integrable Poisson Systems with Asymptotically Decaying Timedependent Perturbation
2015, vol. 20, no. 4, pp. 476485
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.

Cresson J., Wiggins S.
A $\lambda$lemma for Normally Hyperbolic Invariant Manifolds
2015, vol. 20, no. 1, pp. 94108
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 threedegreeoffreedom nearintegrable Hamiltonian systems which satisfy our assumptions.

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. 586600
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 realanalytic 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. 
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. 363373
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.

Waalkens H., Wiggins S.
Geometrical models of the phase space structures governing reaction dynamics
2010, vol. 15, no. 1, pp. 139
Abstract
Hamiltonian dynamical systems possessing equilibria of saddle x center x∙∙∙x center stability type display reactiontype 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 2degreeoffreedom (DoF) systems in the threedimensional 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. 
Rudnev M., Wiggins S.
On a Homoclinic Splitting Problem
2000, vol. 5, no. 2, pp. 227242
Abstract
We study perturbations of Hamiltonian systems of $n+1$ degrees of freedom $(n \geqslant 2)$ in the realanalytic 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 actionangle variables, the splitting is exponentially small.

Rudnev M., Wiggins S.
On a Partially Hyperbolic KAM Theorem
1999, vol. 4, no. 4, pp. 3958
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.
