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A Converse KAM method for 3D vector fields, establishing regions through which pass 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. Minimization of these regions is proposed as a tool to help in the design of plasma confinement devices of tokamak and stellarator type.

We show that the bifurcation diagram proposed in Baesens et al. [Nonlinearity, 31, 2928-2981 (2018)] is the simplest bifurcation diagram for a monotone two-parameter family of vector fields on a torus. We achieve this by finding an example that satisfies all the minimal conditions derived there. We give proofs for all but one of the conditions and present strong numerical evidence for the remaining one., 20 pages, 1 figure

This chapter presents a view of a non-autonomous oscillator as a mapping from input functions of time to a circle of possible solutions (state functions of time). It indicates how this view encompasses chronotaxic systems and enables one, at least conceptually, to understand the extent of synchronisation in networks of oscillators, whether autonomous or not. For the latter a hierarchical aggregation scheme is introduced. The approach is based on the theory of normal hyperbolicity.

A four-parameter family of covariance functions for stationary Gaussian processes is presented. We call it 2Dsys. It corresponds to the general solution of an autonomous second-order linear stochastic differential equation, thus arises naturally from modelling. It covers underdamped and overdamped systems, so it is proposed to use this family when one wishes to decide if a time-series corresponds to stochastically forced damped oscillations or a stochastically forced overdamped system.

The paper presents evidence that Riemann's xi function evaluated at 2 sqrt(E) could be the characteristic function P(E) for the magnetic Laplacian minus 85/16 on a surface of curvature -1 with magnetic field 9/4, a cusp of width 1, a DIrichlet condition at a point, and other conditions not yet determined.

We characterize profit-maximizing operating strategies, over some time horizon [0,T], for an energy store which is trading in an arbitrage market. Our theory allows for leakage, operating inefficiencies, operating constraints and general cost functions. In the special case where the operating cost of a store depends only on its instantaneous power output (or input), we present an algorithm to determine the optimal strategies. A key feature is that this algorithm is localized in time, in the sense that the action of the store at a time t only requires information about electricity prices over some subinterval of time [t,t_k] contained within [t,T]. To introduce more complex storage models, we discuss methods for an example which includes minimum switching times between modes of operation., 19 pages, 1 figure. Submitted to SIAM Journal on Control and Optimization

A transition state for a Hamiltonian system is a closed, invariant, oriented, codimension-2 submanifold of an energy-level that can be spanned by two compact codimension-1 surfaces of unidirectional flux whose union, called a dividing surface, locally separates the energy-level into two components and has no local recrossings. For this to happen robustly to all smooth perturbations, the transition state must be normally hyperbolic. The dividing surface then has locally minimal geometric flux through it, giving an upper bound on the rate of transport in either direction. Transition states diffeomorphic to $\mathbb S^{2m-3}$ are known to exist for energies just above any index-1 critical point of a Hamiltonian of $m$ degrees of freedom, with dividing surfaces $\mathbb S^{2m-2}$. The question addressed here is what qualitative changes in the transition state, and consequently the dividing surface, may occur as the energy or other parameters are varied? We find that there is a class of systems for which the transition state becomes singular and then regains normal hyperbolicity with a change in diffeomorphism class. These are Morse bifurcations. Various examples are considered. Firstly, some simple examples in which transition states connect or disconnect, and the dividing surface may become a torus or other. Then, we show how sequences of Morse bifurcations producing various interesting forms of transition state and dividing surface are present in reacting systems, by considering a hypothetical class of bimolecular reactions in gas phase., 36 pages, final version - some improvements to clarity and notation, fixed typos, added some references

We present a new stochastic framework for studying ship capsize. It is a synthesis of two strands of transition state theory. The first is an extension of deterministic transition state theory to dissipative non-autonomous systems, together with a probability distribution over the forcing functions. The second is stochastic reachability and large deviation theory for transition paths in Markovian systems. In future work we aim to bring these together to make a tool for predicting capsize rate in different stochastic sea states, suggesting control strategies and improving designs., 8 pages, 3 figures, paper for STAB&S 2021 proceedings