David Chillingworth

Discontinuity geometry for an impact oscillator

We use methods of singularity theory to classify the local geometry of the discontinuity set, together with associated local dynamics, for a discrete dynamical system that represents a natural class of oscillator with one degree of freedom impacting against a fixed obstacle. We also include descriptions of the generic transitions that occur in the discontinuity set as the position of the obstacle is smoothly varied. The results can be applied to any choice of restitution law at impact. The analysis provides a general setting for the study of local and global dynamics of discontinuous systems of this type, for example giving a geometric basis for the possible construction of Markov partitions in certain cases.

Approximation of generalized inverses by iterated regularization

Approximations to the Moore-Penrose generalized inverse are obtained via iteration in the application of regularization. Uniform error bounds are obtained for linear operators with closed range. For operators with arbitrary range pointwise error estimates are derived assuming certain smoothness conditions on the data. The stability of the iteration is considered and error bounds are obtained for“noisy”data.

Dynamics of an impact oscillator near a degenerate graze

We give a complete analysis of low-velocity dynamics close to grazing for a generic one degree of freedom impact oscillator. This includes nondegenerate (quadratic) grazing and minimally degenerate (cubic) grazing, corresponding respectively to nondegenerate and degenerate chatter. We also describe the dynamics associated with generic one-parameter bifurcation at a more degenerate (quartic) graze, showing in particular how this gives rise to the often-observed highly convoluted structure in the stable manifolds of chattering orbits. The approach adopted is geometric, using methods from singularity theory.

Geometry and dynamics of a nematic liquid crystal in a uniform shear flow

We study the dynamics of a nematic liquid crystal in a shear flow by employing the gradient of the Landau-de Gennes free-energy function on second-rank tensors, modified by constant and rotational terms. We predict configurations of equilibria and periodic solutions found in numerical simulations and explain certain anomalous nongeneric continua of equilibria. The existence of these continua shows that the model is structurally unstable.

A global genericity theorem for bifurcations in variational problems

Let C be a Banach space, H a Hilbert space, and let F(C,H) be the space of C∞ functions f: C × H → R having Fredholm second derivative with respect to x at each (c, x) ϵ C × H for which Dƒc(x) = 0; here we write ƒc(x) for ƒ(c, x). Say ƒ is of standard type if at all critical points of ƒc it is locally equivalent (as an unfolding) to a quadratic form Q plus an elementary catastrophe on the kernel of Q. It is proved that if f ϵ F (A × B, H) satisfies a certain ‘general position’ condition, and dim B ⩽ 5, then for most a ϵ A the function foϵF(B,H) is of standard type. Using this it is shown that those fϵF(B,H) of standard type form an open dense set in F(B,H) with the Whitney topology. Thus both results are Hilbert-space versions of Thoms theorem for catastrophes in Rn.

Bifurcation from an Orbit of Symmetry

We explore two different approaches to investigating what happens to an orbit of critical points of a symmetric function when the continuous symmetry of the function is broken, and outline an application of the second approach to the traction problem in nonlinear elastostatics.

Generic multiparameter bifurcation from a manifold

The geometry of generic k -parameter bifurcation from an n -manifold is discussed for all values of k , n with particular emphasis on the case n = 2 (the case n = 1 being dealt with in earlier work). Such bifurcations typically arise in the study of equilibrium states of dynamical systems with continuous (for example, spherical or toroidal) symmetry which undergo small symmetry-breaking perturbations, and in the use of Melnikov maps for detecting bifurcations of periodic orbits from resonance. Detailed analysis is given in the interesting case n = 2, k = 3 where the local geometry partly resembles unfolding of a degenerate wavefront or Legendrian collapse.

Integrability of singular distributions on Banach manifolds

One of the key results in the work of the second author ((7), (8)) on integrability of systems of vectorfields is the theorem which relates integrability of a distribution to the concept of homogeneity . In this paper, we show that the homogeneity theorem also applies in an infinite-dimensional context, and this allows us to derive infinite-dimensional versions of several further results in (7) and (8), formulated in terms of distributions. In particular, we are able to express necessary and sufficient conditions for homogeneity in terms of Lie brackets (Theorems 3 and 4) and to characterize integrable real-analytic distributions (Theorem 5). As a corollary to our Theorem 2, we recover the standard Frobenius theorem on the integrability of regular distributions. We also discuss briefly a basic problem which arises in infinite dimensions when we view an integral manifold of an integrable distribution as part of a singular foliation.

Periodic orbits close to grazing for an impact oscillator

We show how the geometric impact surface approach to the dynamics of an impact oscillator provides an immediate visualization of the criteria that determine the existence of an impacting periodic orbit close to grazing. We recover the criteria set out earlier by A. Nordmark and indicate how the geometric setting and singularity geometry may be exploited to yield appropriate criteria in degenerate situations where the Nordmark criteria would not apply.

Molien series and low-degree invariants for a natural action of

We investigate the invariants of the