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In a recent paper we examined a model of an arch bridge with viscous damping subjected to a sinusoidally varying central load. We showed how this yields a useful archetypal oscillator which can be used to study the transition from smooth to discontinuous dynamics as a parameter, α, tends to zero. Decreasing this smoothness parameter (a non-dimensional measure of the span of the arch) changes the smooth load-deflection curve associated with snap-buckling into a discontinuous sawtooth. The smooth snap-buckling curve is not amenable to closed-form theoretical analysis, so we here introduce a piecewise linearization that correctly fits the sawtooth in the limit at α=0. Using a Hamiltonian formulation of this linearization, we derive an analytical expression for the unperturbed homoclinic orbit, and make a Melnikov analysis to detect the homoclinic tangling under the perturbation of damping and driving. Finally, a semi-analytical method is used to examine the full nonlinear dynamics of the perturbed piecewise linear system. A chaotic attractor located at α=0.2 compares extremely well with that exhibited by the original arch model: the topological structures are the same, and Lyapunov exponents (and dimensions) are in good agreement.

The nonlinear axisymmetric post-buckling behaviour of perfect, thin, elastic spherical shells subject to external pressure and their asymmetric bifurcations are characterized, providing results for a structure/loading combination with an exceptionally nonlinear buckling response. Immediately after the onset of buckling, the buckling mode localizes into a dimple at the poles. The relations among the pressure, the dimple amplitude and the change in volume of the shell are determined over a large range of pole deflections. These results allow accurate evaluation of criteria such as the Maxwell condition for which the energies in the unbuckled and buckled states are the same and evaluation of the influences of pressure versus volume-controlled loadings. Non-axisymmetric bifurcation from the axisymmetric state, which occurs deep into the post-buckling regime in the form of multi-lobed dimples, is also established and discussed. This article is part of the themed issue ‘Patterning through instabilities in complex media: theory and applications.’

In this introductory paper, I review the 'visions of the future' articles prepared by top young scientists for the second of the two Christmas 2008 Triennial Issues of Phil. Trans. R. Soc. A, devoted respectively to astronomy and Earth science. Topics covered in the Earth science issue include: trace gases in the atmosphere; dynamics of the Antarctic circumpolar current; a study of the boundary between the Earth's rocky mantle and its iron core; and two studies of volcanoes and their plumes. A final section devoted to ecology and climate covers: the mathematical modelling of plant-soil interactions; the effects of the boreal forests on the Earth's climate; the role of the past palaeoclimate in testing and calibrating today's numerical climate models; and the evaluation of these models including the quantification of their uncertainties.

This brief paper introduces and reviews the 'visions of the future' articles prepared by leading young scientists throughout the world for the first of two Christmas 2008 Triennial issues of Phil. Trans. R. Soc. A, devoted, respectively, to astronomy and Earth science. Contributions in astronomy include the very topical gamma-ray bursts, new ideas on stellar collapse and the unusual atmospheres of synchronized planets orbiting nearby stars.

This paper is the preface to a special Issue of Phil. Trans. R. Soc. A reporting selected proceedings of the international conference marking the 150th anniversary of James Clerk Maxwell's professorial debut at Marischal College, Aberdeen. Following an introduction to Marischal College, a brief historical note summarizes Maxwell's life prior to his entering the college as professor of natural philosophy. The preface provides a short summary of the event and overviews the contributed papers devoted to subjects covering a wide range of Maxwell's research interests and their modern developments. The mixture of review and research papers reflects both the fundamental importance and the diverse applicability of Maxwell's works in electromagnetics, colour science, dynamics and kinetics. Acknowledgements are given to the individuals and bodies who made the conference the success that it was.

Dynamic buckling is addressed for complete elastic spherical shells subject to a rapidly applied step in external pressure. Insights from the perspective of nonlinear dynamics reveal essential mathematical features of the buckling phenomena. To capture the strong buckling imperfection-sensitivity, initial geometric imperfections in the form of an axisymmetric dimple at each pole are introduced. Dynamic buckling under the step pressure is related to the quasi-static buckling pressure. Both loadings produce catastrophic collapse of the shell for conditions in which the pressure is prescribed. Damping plays an important role in dynamic buckling because of the time-dependent nonlinear interaction among modes, particularly the interaction between the spherically symmetric ‘breathing’ mode and the buckling mode. In general, there is not a unique step pressure threshold separating responses associated with buckling from those that do not buckle. Instead, there exists a cascade of buckling thresholds, dependent on the damping and level of imperfection, separating pressures for which buckling occurs from those for which it does not occur. For shells with small and moderately small imperfections, the dynamic step buckling pressure can be substantially below the quasi-static buckling pressure.

This article is an overview of the contributions to the Triennial Issue of Phil. Trans. R. Soc. A published in December, 2005, and also plays the role of a Preface. Devoted to the work of young scientists, the issue covers the fields of astronomy and earth science.

Biomedical researchers regularly stretch and twist single DNA molecules in magnetic tweezer experiments. By making the molecule writhe into a plectoneme (ply) and plotting its load-extension curves, key DNA parameters, such as effective radius, can be estimated. Adding untangling enzymes (topoisomerases) to the DNA's environment, their individual cuts are detected as jumps in extension. Sufficient information is now known about the topoisomerases for us to make good idealizations about their kinematics and mechanics. The novelty of this paper is to study their actions in the context of accurate ply solutions from the theory of elastic rods. To do this, we define an extended rod-plus-tension system that allows us to determine the stored energies from areas in the conventional link versus writhe plane. After a cut, the molecule relaxes dynamically to a new equilibrium state, and often there will be two or more alternative stable configurations onto which it might settle. Knowing the energy levels allows us to identify which states can and cannot be reached over the unstable mountain passes, and which of the accessible states offer the greatest energy relaxation. Strict energy bounds on behaviour are established. This knowledge has medical value because topoisomerase inhibitors, lethal for cells, are used as antibiotics and in chemotherapy for cancer.

This paper provides an informal guide to young researchers in science and engineering as they progress for their first 10 or so years from the time that they first started thinking about doing a PhD. This advice is drawn, with examples and anecdotes, from my own research career which started at the Cambridge Engineering Department in 1958, and progressed through 48 years at University College London to a part-time chair that I now hold in Aberdeen. I hope it may encourage and help tomorrow's scientists on whom the Earth's future very much depends.

Approaching a dangerous bifurcation, from which a dynamical system such as the Earth's climate will jump (tip) to a different state, the current stable state lies within a shrinking basin of attraction. Persistence of the state becomes increasingly precarious in the presence of noisy disturbances. We argue that one needs to extract information about the nonlinear features (a 'softening') of the underlying potential from the time series to judge the probability and timing of tipping. This analysis is the logical next step if one has detected a decrease of the linear decay rate. If there is no discernible trend in the linear analysis, nonlinear softening is even more important in showing the proximity to tipping. After extensive normal-form calibration studies, we check two geological time series from palaeo-climate tipping events for softening of the underlying well. For the ending of the last ice age, where we find no convincing linear precursor, we identify a statistically significant nonlinear softening towards increasing temperature.