We solve 'half' the problem of finding three-dimensional quasisymmetric magnetic fields that do not necessarily satisfy magnetohydrostatic 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.

For an untilted Frenkel–Kontorova chain and any rational p / q , Aubry and Mather proved there are minimising equilibrium states that are left- and right-asymptotic to neighbouring pairs of spatially periodic minimisers of type ( p , q ). They are known as discommensurations (or kinks or fronts), advancing if the right-asymptotic equilibrium is to the right of the left-asymptotic one, retreating otherwise. Following work of Middleton, Floria & Mazo and Baesens & MacKay, there is a threshold tilt F d ( p / q ) ⩾ 0 up to which there continue to be periodic equilibria of type ( p , q ) and above which there is a globally attracting periodically sliding solution in the space of sequences of type ( p , q ). In this paper, we prove that there are values F d ( p / q ± ) of tilt with 0 ⩽ F d ( p / q ± ) ⩽ F d ( p / q ) , generically positive and less than F d ( p / q ) , up to which there continue to be equilibrium advancing or retreating discommensurations, respectively, and such that for F d ( p / q ± ) < F < F d ( p / q ) there are periodically sliding discommensurations, apart perhaps from exceptional cases with both a degenerate type ( p , q ) equilibrium and a degenerate advancing equilibrium discommensuration. We give examples, however, to show that equilibrium and periodically sliding discommensurations may co-exist, both above and below F d ( p / q ± ) , so the case of discommensurations is not as clean as that of periodic configurations. On the way, we prove that F d ( ω ) → F d ( p / q ± ) as ω ↘ p / q or ↗ p / q respectively. Finally, we prove that F d ( p / q ± ) = 0 is equivalent to the existence of a rotational invariant circle consisting of periodic orbits of type ( p , q ) and right-going (respectively left-going) separatrices, for the corresponding twist map on the cylinder.

Numerical tests of volume formulae are presented to efficiently compute the volume enclosed between flux surfaces for integrable 3D vector fields with various degrees of symmetry. In the process, a new case is proposed and tested.

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 degrees 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.

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.

A major achievement of Dewar and coworkers is the SPEC code to construct stepped-pressure equilibria in magnetohydrostatics without axisymmetry. Their existence had been proved by Bruno and Laurence. As part of the procedure of Bruno and Laurence, it is required to solve the Hamilton-Jacobi equation for a magnetic potential on the outside of an interface given the field on the inside and the pressure-jump across the interface. For non-axisymmetric interface, it was understood that solutions with insufficiently irrational rotational transform might not exist, and examples have been given for which there are no solutions at all for large enough pressure-jump. The present paper gives a method to compute regions in the phase space for the pressure-jump Hamiltonian through which no invariant tori pass. The paper also shows how to present the results as regions in the space of pressure-jumps and outer rotational transform for which there is no solution of the Hamilton-Jacobi equation. The method is expected to reach arbitrarily close to the full non-existence region with enough computational work, so what is left over can be relied on to be mostly invariant tori. The paper also brings to attention a class of metrics on tori that are not necessarily axisymmetric yet have integrable geodesic flow. They could give interfaces with solutions for all but finitely many rotational transforms.

The efficiency of a modern economy depends on value-tracking: that market prices of key assets broadly track some underlying value. This can be expected if a sufficient weight of market participants are valuation-based traders, buying and selling an asset when its price is, respectively, below and above their well-informed private valuations. Such tracking will never be perfect, and we propose a natural unit of tracking error, the 'deciblack'. We then use a simple discrete-time model to show how large tracking errors can arise if enough market participants are not valuation-based traders, regardless of how much information the valuation-based traders have. Similarly to Lux [17] and others who study subtly different models, we find a threshold above which value-tracking breaks down without any changes in the underlying value of the asset. We propose an estimator of the tracking error and establish its statistical properties. Because financial markets are increasingly dominated by non-valuation-based traders, assessing how much valuation-based investing is required for reasonable value tracking is of urgent practical interest.

We introduce the Circular Directional Flow Decomposition (CDFD), a new framework for analyzing circularity in weighted directed networks. CDFD separates flow into two components: a circular (divergence-free) component and an acyclic component that carries all nett directional flow. This yields a normalized circularity index between 0 (fully acyclic) and 1 (for networks formed solely by the superposition of cycles), with the complement measuring directionality. This index captures the proportion of flow involved in cycles, and admits a range of interpretations - such as system closure, feedback, weighted strong connectivity, structural redundancy, or inefficiency. Although the decomposition is generally non-unique, we show that the set of all decompositions forms a well-structured geometric space with favourable topological properties. Within this space, we highlight two benchmark decompositions aligned with distinct analytical goals: the maximum circularity solution, which minimizes nett flow, and the Balanced Flow Forwarding (BFF) solution, a unique, locally computable decomposition that distributes circular flow across all feasible cycles in proportion to the original network structure. We demonstrate the interpretive value and computational tractability of both decompositions on synthetic and empirical networks. They outperform existing circularity metrics in detecting meaningful structural variation. The decomposition also enables structural analysis - such as mapping the distribution of cyclic flow - and supports practical applications that require explicit flow allocation or routing, including multilateral netting and efficient transport.

It is proved that for families of stochastic operators on a countable tensor product, depending smoothly on parameters, any spectral projection persists smoothly, where smoothness is defined using norms based on ideas of Dobrushin. A rigorous perturbation theory for families of stochastic operators with spectral gap is thereby created. It is illustrated by deriving an effective slow two-state dynamics for a three-state probabilistic cellular automaton.

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.

We present an example of a monotone two-parameter family of vector fields on a torus whose bifurcation diagram we demonstrate to be in the class of 'simplest' diagrams proposed by Baesens and MacKay (2018 Nonlinearity 31 2928-81). This shows that the proposed class is realisable.

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. 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 with some simple examples.

A converse KAM method for 3D vector fields, establishing regions through which passes no invariant 2-tori transverse to a given direction field, is tested on some helical perturbations of an axisymmetric magnetic field in toroidal geometry. It finds regions corresponding to magnetic islands and chaos for the fieldline flow. The minimization of these regions is proposed as a tool to help in the design of plasma confinement devices of tokamak and stellarator type.

In this paper we apply Aubry-Mather theory for equilibria of 1D Hamiltonian lattice systems and the theory of invariant ordered circles to investigate the depinning transition of travelling waves for particle chains. Assume A < B are two critical values such that the particle chain has three homogeneous equilib-ria if the driving force F E (A, B). It is already known that there exist transition thresholds F-c ?F+ (c) of the driving force such that the particle chain has station-ary fronts but no travelling fronts for F-c ?F ? F+c and travelling fronts but no stationary fronts if A < F < F- (c )or F+ (c) < F < B. The novelty of our approach is that we prove the transition threshold F+ c (F-c) coincides with the upper (lower) limit of the upper (lower) depinning force as the rotation number tends to zero from the right. Based on this conclusion, we demonstrate that when the driving force F E (F-(c), F-c(+)), besides stationary fronts there are various kinds of equilibria with rotation numbers close to zero such that the spatial shift map has positive topological entropy on the set of equilibria. Furthermore, we give a necessary and sufficient condition for the absence of propagation failure, i.e. F-( c) = F-c(+) , in terms of a minimal foliation. Finally we show that F- c(+/-) are continuous with respect to potential functions in C-1 topology.

For Schrodinger operators with suitable 1D potentials, focussing particularly on those that go to infinity at infinity, a characteristic function is constructed, via shooting functions. It is proved to be entire and its zeroes to be the eigenvalues.

A method to establish regions of phase space through which pass no invariant tori transverse to a given direction field is applied to the planar circular restricted three-body problem. Implications for the location of stable orbits for planets around a binary star are deduced. It is expected that lessons learnt from this problem will be useful for applications of the method to other contexts such as flux surfaces for magnetic fields, guiding centre motion in magnetic fields, and classical models of chemical reaction dynamics.

Abstract This corrigendum corrects a mistake in the paper (Baesens and MacKay 2018 Nonlinearity 31 2928–81) and the consequent conclusions. The corrected conclusions are confirmed by numerically computed bifurcation diagrams.

This corrigendum corrects a mistake in the paper (Baesens and MacKay 2018 Nonlinearity 31 2928-81) and the consequent conclusions. The corrected conclusions are confirmed by numerically computed bifurcation diagrams.