Anne C. Skeldon

Stability results for steady, spatially periodic planforms

Equivariant bifurcation theory has been used extensively to study pattern formation via symmetry-breaking steady-state bifurcation in various physical systems modelled by E(2)-equivariant partial differential equations. Much attention has been focused on solutions that are doubly periodic with respect to a square or hexagonal lattice, for which the bifurcation problem can be restricted to a finite-dimensional centre manifold. Previous studies have used four- and six-dimensional representations for the square and hexagonal lattice symmetry groups respectively, which in turn allows the relative stability of squares and rolls or hexagons and rolls to be determined. Here we consider the countably infinite set of eight- and 12-dimensional irreducible representations for the square and hexagonal cases, respectively. This extends earlier relative stability results to include a greater variety of bifurcating planforms, and also allows the stability of rolls, squares and hexagons to be established to a countably infinite set of perturbations. In each case we derive the Taylor expansion of the equivariant bifurcation problem and compute the linear, orbital stability of those solution branches guaranteed to exist by the equivariant branching lemma. In both cases we find that many of the stability results are established at cubic order in the Taylor expansion, although to completely determine the stability of certain states, higher-order terms are required. For the hexagonal lattice, all of the solution branches guaranteed by the equivariant branching lemma are, generically, unstable due to the presence of a quadratic term in the Taylor expansion. For this reason we consider two special cases: the degenerate bifurcation problem that is obtained by setting the coefficient of the quadratic term to zero, and the bifurcation problem when an extra reflection symmetry is present.

Free convection in liquid gallium

Free convection in liquid metals is of significant practical interest to the crystal-growing community since the adverse effects of convective instabilities in the melt phase can be frozen into the solid product. Here, we present the results of a combined numerical and experimental study of steady convective flows in a sample of liquid gallium which is heated at one end and cooled at the other. Experimental measurements of temperature distributions in the flow are compared with the standard Hadley-cell solution and with the numerical results obtained from a two-dimensional model. Excellent quantitative agreement is found between all three for low Grashof numbers but a systematic divergence between the results is seen as this parameter is increased.

A study of two novel self-exciting single-disk homopolar dynamos: Theory

We investigate the novel set of nonlinear ordinary differential equations x = x (y —1) —βz, y = α(1 — x2)— ky and z = x-λz, where x=dx/dτ, etc. They govern the behaviour of two (mathematically equivalent) self-exciting homopolar dynamo systems, each comprising a Faraday disk and coil arrangement, the first with a capacitor and the second with a motor connected in series with the coil, where in each case the applied couple G that drives the disk (of moment of inertia A) into rotation with angular speed Ω(t) is assumed steady. The independent variable τ denotes time t measured in units of L/Rs, where L is the self-inductance of the system and R the total series resistance. The dependent variable x(τ) is the electric current I(t) generated in the system measured in units of (G/M)1/2 A, where 2πM is the mutual inductance between the disk and coil, and y(τ) corresponds to Ω(t) measured in units R/M rad s-1 . In the case of the series capacitor (of capacitance C), the third independent variable z(τ) is the charge Q(t) on the capacitor measured in units (G/M)1/2(L/R) C; in the case of the series motor, z(τ) is the angular speed ω(t) with which the armature of the motor is driven into rotation by the torque HI(t) due to the current I(t) passing through it, measured in units (L/R)(M/G)1/2(H/B)rad s-1 , where B is the moment of inertia of the armature. Common to both systems are the dimensionless parameters α = GLM/R2A and K = KL/RA, where K is the coefficient of (linear) mechanical friction in the disk. In the case of the capacitor, β = L/CR2 and λ = L/RrC where r is the leakage resistance of the capacitor; in the case of the motor, β = H2L/R2B and λ = DL/RB where D is the coefficient of (linear) mechanical friction in the motor. The behaviour of the system, including its sensitivity to initial conditions, depends on the four parameters (α, β, κ, λ), the least interesting case being when the system fails to function as a self-exciting dynamo capable of amplifying a small adventitious electric current because α/κ At is not large enough for motional induction to overcome ohmic dissipation. Otherwise, i.e. where α/κ exceeds a critical value dependent on β and λ, dynamo action occurs in which the detailed time dependence of the current x(τ) and of the other variables y(τ) and z(τ) depends critically on the exact values of (α, β, κ, λ), and in some cases also on the initial conditions. In the simplest cases, x(τ) tends to solutions which (apart from the sign of x(τ), which depends of the sign of the initial disturbance) are either independent of τ or vary harmonically with τ. At other values of (α, β, κ, λ), multiple solutions are found, some of which are periodic (but non-harmonic), including square x(τ) and saw-tooth y(τ) and z(τ) waveforms, and others chaotic. A full elucidation of this behaviour will require extensive numerical studies over wide ranges of all these parameters, but bifurcation theory applied to the stability or otherwise of the equilibrium solutions (x0,y0, z0) = (0, α/κ, 0) and (±[1 — (κ/α)(1 + β/λ)]1/2, 1 + β/λ, x0/λ) provides theoretical guidance. It shows in particular the usefulness of a regime diagram in parameter space of the first quadrant of the (β, α/κ) plane where there is one line α/κ = 1 + β/λ where symmetry breaking bifurcations occur and parts of two lines α/κ = 1 + λ and α/κ = [(2β - κλ - λ2)/2(κ-β/λ) + 3β/2λ + 1] upon which Hopf bifurcations occur, all meeting at the point (β, α/κ) = (λ2, 1 + λ) of the Takens-Bogdanov ‘double-zero eigenvalue’ type, with reflectional symmetry. The equations governing self-exciting homopolar dynamos can be used as the basis of nonlinear low-dimensional analogues in the study of the temporal behaviour of certain phenomena of interest in geophysical fluid dynamics. These include the main geomagnetic field produced by self-exciting magnetohydrodynamic (MHD) dynamo action in the Earth’s liquid metallic core and the ‘El Niño-Southern Oscillation’ of the Earth’s atmosphere-ocean system (considerations of which prompted the present study), which has certain characteristics resembling those found in nonlinear relaxation oscillators and is produced by complex global-scale interactions between the atmosphere and oceans. Geophysical implications of the findings of the present study of simple (but not over-simplified) physically realistic dynamos will be discussed elsewhere, in the context of the further computational investigations needed to elucidate more fully the rich and complex behaviour indicated by the results obtained to date.

Dynamics of a parametrically excited double pendulum

Abstract The 2:2 mode interaction of a parametrically excited double pendulum is explored in the excitation frequency/excitation amplitude plane. To determine the bifurcation structure at small amplitudes of oscillation, the method of averaging combined with centre manifold reduction is used. The full equations are solved numerically to extend the bifurcation set to larger amplitudes of response. Numerical centre manifold reduction is employed to derive two maps, valid near two multiple bifurcation points which organise the dynamical phenomena of the mode interaction region. Iteration of these maps shows the existence of global bifurcations. These results are discussed in the light of numerical integrations which show that a whole range of interesting behaviour occurs including torus doubling, torus ‘gluing’ and chaos.

Mathematical Models for Sleep-Wake Dynamics: Comparison of the Two-Process Model and a Mutual Inhibition Neuronal Model

Sleep is essential for the maintenance of the brain and the body, yet many features of sleep are poorly understood and mathematical models are an important tool for probing proposed biological mechanisms. The most well-known mathematical model of sleep regulation, the two-process model, models the sleep-wake cycle by two oscillators: a circadian oscillator and a homeostatic oscillator. An alternative, more recent, model considers the mutual inhibition of sleep promoting neurons and the ascending arousal system regulated by homeostatic and circadian processes. Here we show there are fundamental similarities between these two models. The implications are illustrated with two important sleep-wake phenomena. Firstly, we show that in the two-process model, transitions between different numbers of daily sleep episodes can be classified as grazing bifurcations. This provides the theoretical underpinning for numerical results showing that the sleep patterns of many mammals can be explained by the mutual inhibition model. Secondly, we show that when sleep deprivation disrupts the sleep-wake cycle, ostensibly different measures of sleepiness in the two models are closely related. The demonstration of the mathematical similarities of the two models is valuable because not only does it allow some features of the two-process model to be interpreted physiologically but it also means that knowledge gained from study of the two-process model can be used to inform understanding of the behaviour of the mutual inhibition model. This is important because the mutual inhibition model and its extensions are increasingly being used as a tool to understand a diverse range of sleep-wake phenomena such as the design of optimal shift-patterns, yet the values it uses for parameters associated with the circadian and homeostatic processes are very different from those that have been experimentally measured in the context of the two-process model.

Two-frequency forced Faraday waves: weakly damped modes and pattern selection

Abstract Recent experiments [A. Kudrolli, B. Pier, J.P. Gollub, Physica D 123 (1998) 99–111] on two-frequency parametrically excited surface waves produced an intriguing “superlattice” wave pattern near a codimension-two bifurcation point where both subharmonic and harmonic waves onset simultaneously, but with different spatial wave numbers. The superlattice pattern is synchronous with the forcing, spatially periodic on a large hexagonal lattice, and exhibits small-scale triangular structure. Similar patterns have been shown to exist as primary solution branches of a generic 12-dimensional D 6 + T 2 -equivariant bifurcation problem, and may be stable if the nonlinear coefficients of the bifurcation problem satisfy certain inequalities [M. Silber, M.R.E. Proctor, Phys. Rev. Lett. 81 (1998) 2450–2453]. Here we use the spatial and temporal symmetries of the problem to argue that weakly damped harmonic waves may be critical to understanding the stabilization of this pattern in the Faraday system. We illustrate this mechanism by considering the equations developed by Zhang and Vinals [J. Fluid Mech. 336 (1997) 301–330] for small amplitude, weakly damped surface waves on a semi-infinite fluid layer. We compute the relevant nonlinear coefficients in the bifurcation equations describing the onset of patterns for excitation frequency ratios of 2 3 and 6 7 . For the 2 3 case, we show that there is a fundamental difference in the pattern selection problems for subharmonic and harmonic instabilities near the codimension-two point. Also, we find that the 6 7 case is significantly different from the 2 3 case due to the presence of additional weakly damped harmonic modes. These additional harmonic modes can result in a stabilization of the superpatterns.

Mode interaction in a double pendulum

Abstract The results of a theoretical and experimental study of a parametrically excited double pendulum with Z 2 ⊕ Z 2 symmetry are presented. Local analysis of the equations at multiple bifurcation points demonstrates the existence of global bifurcations. These suggest the presence of torus-doubling and chaos, both of which are observed experimentally and numerically.

Modelling changes in sleep timing and duration across the lifespan: Changes in circadian rhythmicity or sleep homeostasis?

Sleep changes across the lifespan, with a delay in sleep timing and a reduction in slow wave sleep seen in adolescence, followed by further reductions in slow wave sleep but a gradual drift to earlier timing during healthy ageing. The mechanisms underlying changes in sleep timing are unclear: are they primarily related to changes in circadian processes, or to a reduction in the neural activity dependent build up of homeostatic sleep pressure during wake, or both? We review existing studies of age-related changes to sleep and explore how mathematical models can explain observed changes. Model simulations show that typical changes in sleep timing and duration, from adolesence to old age, can be understood in two ways: either as a consequence of a simultaneous reduction in the amplitude of the circadian wake-propensity rhythm and the neural activity dependent build-up of homeostatic sleep pressure during wake; or as a consequence of reduced homeostatic sleep pressure alone. A reduction in the homeostatic pressure also explains greater vulnerability of sleep to disruption and reduced daytime sleep-propensity in healthy ageing. This review highlights the important role of sleep homeostasis in sleep timing. It shows that the same phenotypic response may have multiple underlying causes, and identifies aspects of sleep to target to correct delayed sleep in adolescents and advanced sleep in later life.

Pattern selection for faraday waves in an incompressible viscous fluid

When a layer of fluid is oscillated up and down with a sufficiently large amplitude, patterns form on the surface, a phenomenon first observed by Faraday. A wide variety of such patterns have been observed from regular squares and hexagons to superlattice and quasipatterns and more exotic patterns such as oscillons. Previous work has investigated the mechanisms of pattern selection using the tools of symmetry and bifurcation theory. The hypotheses produced by these generic arguments have been tested against an equation derived by Zhang and Vinals in the weakly viscous and large depth limit. However, in contrast, many of the experiments use shallow viscous layers of fluid to counteract the presence of high frequency weakly damped modes that can make patterns hard to observe. Here we develop a weakly nonlinear analysis of the full Navier–Stokes equations for the two‐frequency excitation Faraday experiment. The problem is formulated for general depth, although results are presented only for the infinite dept...

On a codimension-three bifurcation arising in a simple dynamo model

Abstract In this paper we investigate the dynamics associated with a degenerate codimension-two Takens-Bogdanov bifurcation which arises in a recently derived model for self-exciting dynamo action introduced by Hide et al. [R. Hide, A.C. Skeldon, D.J. Acheson, A study of two novel self-exciting single-disk homopolar dynamos: theory, Proc. R. Soc. Lond. A 452 (1996) 1369–1395]. The general unfolding of such a codimension-three bifurcation has already been discussed in an abstract setting by Li and Rousseau [Codimension-2 symmetric homoclinic bifurcations and application to 1 : 2 resonance, Can J. Math. 42 (1990) 191–212]. Here we describe the unfolding scenario in the context of the dynamo problem. In particular we compare the behaviour predicted by the normal form analysis with a bifurcation study of the full dynamo equations in the neighbourhood of the codimension-three point.