Search

We solve "half" the problem of finding three-dimensional quasisymmetric magnetic fields that do not necessarily satisfy force balance. This involves determining which hidden symmetries are admissible as quasisymmetries, and then showing explicitly how to construct quasisymmetric magnetic fields given an admissible symmetry. The admissibility conditions take the form of a system of overdetermined nonlinear partial differential equations involving second derivatives of the symmetry's infinitesimal generator., Comment: 18 pages

Escape from a potential well occurs in a wide variety of physical systems from chemical reactions to ship capsize. In these situations escape often occurs by passage over a normally hyperbolic submanifold (NHS). This paper describes the computational implementation of an algorithm to identify the NHS of a two degree of freedom model for ship motion under the influence of damping and aperiodic forcing. Additionally we demonstrate how the stable manifolds of these submanifolds can be used to classify initial ship states as safe or unsafe., Comment: Referee version (double line spacing) 51 pages, 15 figures, submitted to Chaos

In a magnetic field, transitions between classes of guiding-centre motion can lead to cross-field diffusion and escape. We say a magnetic field is isodrastic if guiding centres make no transitions between classes of motion. Therefore, this is an important ideal for enhancing confinement. First, we present a weak formulation, based on the longitudinal adiabatic invariant, generalising omnigenity. To demonstrate that isodrasticity is strictly more general than omnigenity, we construct weakly isodrastic mirror fields that are not omnigenous. Then we present a strong formulation that is exact for guiding-centre motion. We develop a first-order treatment of the strong version via a Melnikov function and show that it recovers the weak version. The theory provides quantification of deviations from isodrasticity that can be used as objective functions in optimal design. The theory is illustrated on some simple examples., Comment: 74 pages, 30 Figures, v2.0

For exact area-preserving twist maps, curves were constructed through the gaps of cantori in \cite{MMP84}, which were conjectured to have minimal flux subject to passing through the points of the cantorus. It was pointed out by \cite{Pol} that these curves do {\em not} have minimal flux if there coexists a rotational invariant circle of a different rotation number, but if hyperbolic they do have {\em locally} minimal flux even without the constraint of passing through the points of the cantorus. Following the criterion of \cite{M94} for surfaces of locally minimal flux for 3D volume-preserving flows, I revisit this result and show that in general the analogous curves through the points of rotationally-ordered periodic orbits or their heteroclinic orbits do {\em not} have locally minimal flux. Along the way, various questions are posed. Some results for more degrees of freedom are summarised., Comment: v1 dates from 1 Dec 2019; this v2 dates from 22 July 2020

It is proved that two useful and apparently different metrics on the set of Borel probabilities on countable products of Polish spaces of bounded diameters are equal. This paves the way for advances in their computation.

The structure of static MHD equilibria that admit continuous families of Euclidean symmetries is well understood. Such field configurations are governed by the classical Grad-Shafranov equation, which is a single elliptic PDE in two space dimensions. By revealing a hidden symmetry, we show that in fact all smooth solutions of the equilibrium equations with non-vanishing pressure gradients away from the magnetic axis satisfy a generalization of the Grad-Shafranov equation. In contrast to solutions of the classical Grad-Shafranov equation, solutions of he generalized equation are not automatically equilibria, but instead only satisfy force balance averaged over the one-parameter hidden symmetry. We then explain how the generalized Grad-Shafranov equation can be used to reformulate the problem of finding exact three-dimensional smooth solutions of the equilibrium equations as finding an optimal volume-preserving symmetry., Comment: 11 pages

The trophic levels of nodes in directed networks can reveal their functional properties. Moreover, the trophic coherence of a network, defined in terms of trophic levels, is related to properties such as cycle structure, stability and percolation. The standard definition of trophic levels, however, borrowed from ecology, suffers from drawbacks such as requiring source nodes, which limit its applicability. Here we propose a simple new definition of trophic level that can be computed on any directed network. We demonstrate how the method can identify node function in examples including ecosystems, supply chain networks, gene expression, and global language networks. We also explore how trophic levels and coherence relate to other topological properties, such as non-normality and cycle structure, and show that our method reveals the extent to which the edges in a directed network are aligned in a global direction.

We define Persistent Mutual Information (PMI) as the Mutual (Shannon) Information between the past history of a system and its evolution significantly later in the future. This quantifies how much past observations enable long term prediction, which we propose as the primary signature of (Strong) Emergent Behaviour. The key feature of our definition of PMI is the omission of an interval of 'present' time, so that the mutual information between close times is excluded: this renders PMI robust to superposed noise or chaotic behaviour or graininess of data, distinguishing it from a range of established Complexity Measures. For the logistic map we compare predicted with measured long-time PMI data. We show that measured PMI data captures not just the period doubling cascade but also the associated cascade of banded chaos, without confusion by the overlayer of chaotic decoration. We find that the standard map has apparently infinite PMI, but with well defined fractal scaling which we can interpret in terms of the relative information codimension. Whilst our main focus is in terms of PMI over time, we can also apply the idea to PMI across space in spatially-extended systems as a generalisation of the notion of ordered phases., Comment: 14 pages, 5 figures, revised version as resubmitted to Advances in Complex Systems 1 April 2010

We consider a sequences of symplectic twist maps corresponding to Frenkel Kontorova variational functional. For such a sequence without conjugate points we construct an invariant Lagrangian subbundle. Using the ideas by E.Hopf from Riemannian geometry we show that absence of conjugate points is a rare property for this class of maps., Comment: 12 pages with no fgures

We construct an approximate renormalization transformation for Hamiltonian systems with three degrees of freedom in order to study the break-up of invariant tori with three incommensurate frequencies which belong to the cubic field $Q(\tau)$, where $\tau^3+\tau^2-2\tau-1=0$. This renormalization has two fixed points~: a stable one and a hyperbolic one with a codimension one stable manifold. We compute the associated critical exponents that characterize the universality class for the break-up of the invariant tori we consider., Comment: 5 pages, REVTeX

We consider in this paper elliptic equations which are perturbations of Laplace's equation by a compactly supported potential. We show that in dimension greater than three for a wide class of potentials all the solutions are globally minimising. However, in dimension two the situation is different. We show that for radially symmetric potentials there always exist solutions which are not locally minimal unless the potential vanishes identically. We discuss the relations of this with the so-called Hopf rigidity phenomenon., Comment: 11 pages, no figures

Discrete breathers are time-periodic, spatially localized solutions of equations of motion for classical degrees of freedom interacting on a lattice. They come in one-parameter families. We report on studies of energy properties of breather families in one-, two- and three-dimensional lattices. We show that breather energies have a positive lower bound if the lattice dimension of a given nonlinear lattice is greater than or equal to a certain critical value. These findings could be important for the experimental detection of discrete breathers., Comment: 10 pages, LaTeX, 4 figures (ps), Physical Review Letters, in print

We study the stationary states of networks consisting of weakly coupled bistable units. We prove the existence of a high multiplicity of stable steady states in networks with very general inter-unit dynamics. We present a method for estimating the critical coupling strength below which these stationary states persist in the network. In some cases, the presence of time-independent localized states in the system can be regarded as a `propagation failure' phenomenon. We analyse this type of behaviour in the case of diffusive networks whose elements are described by one or two variables and give concrete examples., Comment: 23 pages, .dvi format in uuencode file. No figures. To appear in Physica D