The global standard for the Llandovery Series (early Silurian) in central Wales is re-assessed in the light of detailed geological surveying, biostratigraphical sampling and a rigorous examination of published datasets. A new sedimentary and biostratigraphical architecture is presented. Key graptolite, brachiopod, acritarch and, for the first time, chitinozoan assemblages are critically assessed. Upper Hirnantian to Aeronian strata record events that followed the Late Ordovician glacial maximum and comprise a series of progradational sequences bounded by flooding surfaces, but inferred still to be glacioeustatic in origin. Significant faunal renewals associated with many of the flooding levels underpin their potential for international recognition. Compound non-sequences are a feature of proximal parts of the system where erosion associated with fault footwall uplift was an important process. Extensive slump sheets contribute to further stratal loss and displacement in distal facies. A re-assessment of the Aeronian Stage GSSP reveals shortcomings with the biostratigraphical criteria used in its selection. Telychian portions of the succession display the disrupting effects of intra-Wenlock synsedimentary sliding; hence the relevance of key published fossil assemblages and the criteria used to erect the stage GSSP are undermined. However, the Llandovery area remains one of the best studied early Silurian successions in the world. This, together with regional considerations, supports the retention of the series standard in mid Wales where the contiguous deep-water basinal succession affords internationally cited exposure of richly graptolitic facies for the whole series and, significantly, for the post-sedgwickii Biozone interval.

In [8] we studied equivariant bifurcation problems with a symmetry group acting on parameters, from the point of view of singularity theory. We followed the now classical theory originated by Damon [5], using the ideas presented in [5, 13, 14]. We adapted general results about unfoldings, the algebraic characterization of finite determinacy, and the recognition problems, to multiparameter bifurcation problems f(x, λ)=0 with ‘diagonal’ symmetry on both the state variables and on the bifurcation parameters. More precisely, such bifurcation problems satisfy the condition f(γx, γλ)=γf(x, λ) for all γ∈Γ, where Γ is a compact Lie group

In this paper we attack the same problem from a different angle: the path formulation. This idea can be traced back to the first papers of Mather [17] and Martinet [15, 16]. It was used explicitly in Golubitsky and Schaeffer [12] (see also their earlier paper [11]) as a way of relating bifurcation problems in one state variable without symmetry to a miniversal unfolding in the sense of catastrophe theory. At that time the techniques of singularity theory were not powerful enough to handle the full power of the idea efficiently – either in theory or in computational practice. This is why the path formulation was abandoned in favour of contact equivalence with distinguished parameters, as developed in Golubitsky and Schaeffer [12]. Considerable progress has been made since then; for example Montaldi and Mond [19] use the path formulation to apply the idea of [Kscr ]V-equivalence introduced by Damon [6] to equivariant bifurcation theory. Bridges and Furter [3] studied equivariant gradient bifurcation problems using the path formulation, and defined an equivalence relation in the space of paths and their unfoldings that respects contact equivalence of the gradients. Here we describe an algebraic approach to the path formulation that has the advantage of organizing the classification of normal forms. Moreover, it minimizes the calculation involved in obtaining the normal forms (compare with the classical framework in Furter et al. [8]). The geometric approach to the path formulation using [Kscr ]V-equivalence is still open in the context of a symmetry group acting diagonally on parameters.

The study of equivariant bifurcation problems via singularity theory (Golubitsky and Schaeffer[8], Golubitsky, Stewart and Schaeffer[9]) has been mainly concerned with models exhibiting spontaneous symmetry-breaking. The solutions of such bifurcation problems lose symmetry as the parameters vary, but the equations that they satisfy retain the same symmetry throughout.