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Eye movement instabilities and nystagmus can be predicted by a nonlinear dynamics model of the saccadic system
No full textThe study of eye movements and oculomotor disorders has, for four decades, greatly benefitted from the application of control theoretic concepts. This paper is an example of a complementary approach based on the theory of nonlinear dynamical systems. Recently, a nonlinear dynamics model of the saccadic system was developed, comprising a symmetric piecewise-smooth system of six first-order autonomous ordinary differential equations. A preliminary numerical investigation of the model revealed that in addition to generating normal saccades, it could also simulate inaccurate saccades, and the oscillatory instability known as congenital nystagmus (CN). By varying the parameters of the model, several types of CN oscillations were produced, including jerk, bidirectional jerk and pendular nystagmus.
The aim of this study was to investigate the bifurcations and attractors of the model, in order to obtain a classification of the simulated oculomotor behaviours. The application of standard stability analysis techniques, together with numerical work, revealed that the equations have a rich bifurcation structure. In addition to Hopf, homoclinic and saddlenode bifurcations organised by a Takens-Bogdanov point, the equations can undergo nonsmooth pitchfork bifurcations and nonsmooth gluing bifurcations. Evidence was also found for the existence of Hopf-initiated canards.
The simulated jerk CN waveforms were found to correspond to a pair of post-canard symmetry-related limit cycles, which exist in regions of parameter space where the equations are a slow-fast system. The slow and fast phases of the simulated oscillations were attributed to the geometry of the corresponding slow manifold. The simulated bidirectional jerk and pendular waveforms were attributed to a symmetry invariant limit cycle produced by the gluing of the asymmetric cycles. In contrast to control models of the oculomotor system, the bifurcation analysis places clear restrictions on which kinds of behaviour are likely to be associated with each other in parameter space, enabling predictions to be made regarding the possible changes in the oscillation type that may be observed upon changing the model parameters. The analysis suggests that CN is one of a range of oculomotor disorders associated with a pathological saccadic braking signal, and that jerk and pendular nystagmus are the most probable oscillatory instabilities. Additionally, the transition from jerk CN to bidirectional jerk and pendular nystagmus observed experimentally when the gaze angle or attention level is changed is attributed to a gluing bifurcation. This suggests the possibility of manipulating the waveforms of subjects with jerk CN experimentally to produce waveforms with an extended foveation period, thereby improving visual resolution
Dimensionality Reduction Using Secant-Based Projection Methods: The Induced Dynamics in Projected Systems
No full textIn previous papers we have developed an approach to the data reduction problem which is based on a well-known, constructive proof of Whitney’s embedding theorem [Broomhead, D. S. and Kirby, M., SIAM Journal of Applied Mathematics 60(6), 2000, 2114–2142; Broomhead, D. S. and Kirby, M., Neural Computation 13, 2001, 2595–2616]. This approach involves picking projections of the high-dimensional system which are optimised in the sense that they are easy to invert. This is done by considering the effect of the projections on the set of unit secants constructed from the data. In the present paper we discuss the implications of this idea in the case that the high-dimensional data is generated by a dynamical system. We ask if the existence of an easily invertible projection leads to practical methods for the construction of an equivalent, low-dimensional dynamical system. The paper consists of a review of the secant-based projection method and simple methods for finding good representations of the (nonlinear) inverse of the projections. We then discuss two variants of a way to find the dynamical system induced by a projection which lead to quite distinct numerical approximations. One of these is developed further as we describe various ways in which knowledge of the full dynamical system can be incorporated into the approximate projected system. The ideas of the paper are illustrated in some more or less simple examples, which range from a simple system of nonlinear ODEs which have an attracting limit cycle, to low-dimensional solutions of the Kuramoto–Sivashinsky equation which need many Galerkin modes for their description
Coupled thermoelasticity in a composite half-space
Get PDFThe problem of fully coupled thermoelasticity in a composite half-space is considered where the composite has variations in its physical properties in one direction only. The resulting one-dimensional problem thus depends on the so-called microscale of the composite. Homogenization of the fully coupled theory provides the leading order system of coupled equations (independent of the microscale) together with the effective physical properties of the thermoelastic medium. In particular the effective coupling parameter δ is found and it is shown to exhibit rather interesting properties; for a range of volume fractions in two-phase composites it is shown that δ lies below the corresponding coupling parameter for a homogeneous material made up of either phase. Transient boundary value problems of the homogenized system are then solved and compared with the classical problem of a homogeneous half-space. The magnitude of resulting discontinuities in field variables and their derivatives are found and their dependence on the effective coupling parameter is exhibited
On the asymptotic properties of multivariate sample autocovariances
No full textWe show that if a process can be obtained by filtering an autoregressive process, then the asymptotic distribution of sample autocovariances of the former is the same as the asymptotic distribution of linear combinations of sample autocovariances of the latter. This result is used to show that for small lags the sample autocovariances of the filtered process have the same asymptotic distribution as estimators utilizing more information (observations on the associated autoregression process and knowledge of the parameters of the filter). In particular, for a Gaussian ARMA process the first few sample autocovariances are jointly asymptotically efficient
Computational modelling with functional differential equations: identification, selection and sensitivity
No full textMathematical models based upon certain types of differential equations, functional differential equations, or systems of such equations, are often employed to represent the dynamics of natural, in particular biological, phenomena. We present some of the principles underlying the choice of a methodology (based on observational data) for the computational identification of, and discrimination between, quantitatively consistent models, using scientifically meaningful parameters.
We propose that a computational approach is essential for obtaining meaningful models. For example, it permits the choice of realistic models incorporating a time-lag which is entirely natural from the scientific perspective. The time-lag is a feature that can permit a close reconciliation between models incorporating computed parameter values and observations. Exploiting the link between information theory, maximum likelihood, and weighted least squares, and with distributional assumptions on the data errors, we may construct an appropriate objective function to be minimized computationally. The minimizer is sought over a set of parameters (which may include the time-lag) that define the model. Each evaluation of the objective function requires the computational solution of the parametrized equations defining the model. To select a parametrized model, from amongst a family or hierarchy of possible best-fit models, we are able to employ certain indicators based on information-theoretic criteria. We can evaluate confidence intervals for the parameters, and a sensitivity analysis provides an expression for an information matrix, and feedback on the covariances of the parameters in relation to the best fit. This gives a firm basis for any simplification of the model (e.g., by omitting a parameter)
The envelope of lines meeting a fixed line and tangent to two spheres
No full textWe study the set of lines that meet a fixed line and are tangent to two spheres and classify the configurations consisting of a single line and three spheres for which there are infinitely many lines tangent to the three spheres that also meet the given line. All such configurations are degenerate. The path to this result involves the interplay of some beautiful and intricate geometry of real surfaces in 3-space, complex projective algebraic geometry, explicit computation and graphics
Diffraction of flexural waves by cracks in orthotropic thin elastic plates. I - formal solution
No full textThe problem of flexural wave diffraction by a semi-infinite crack in an infinite orthotropic thin plate is considered. Such models have application to the ultrasonic non-destructive inspection of thin components, such as aeroplane wings. For simplicity, the plate is modelled using Kirchhoff theory, and the crack is chosen to be aligned along one of the principal directions of material orthotropy. For incident plane waves, an exact analytical expression for the scattered field is derived by means of the Wiener–Hopf technique. In this model problem, the Wiener–Hopf kernel is scalar and its factorization is expressed in terms of simple, definite, non-singular contour integrals. A detailed numerical evaluation of the solution will be provided in the second part of this work
A porous medium model for vegetation in bushfires
Get PDFThe vegetation that burns in bushfires consists of water, pyrolysable material that can ultimately burn as a gas, solid combustible material (char) that is capable only of surface burning and unburnable material (ash). The vegetation is also a porous medium in which air, water vapour pyrolysis products and combustion products flow past solid material. This flow is strongly influenced by and interacts with the flow of air above the layer of vegetation. As unburnt vegetation, the material is all contained in objects of different size and shape, such as leaves, stalks, branches and trunks, all exhibiting different rates at which water and pyrolysis products can be generated through external heating, making some components much more susceptible to burning in a brief period while others would require longer time-scales; such material can often survive a fire. This article presents a sectional model for the vegetation in which different sizes of material are represented separately
Symmetric Hamiltonian Bifurcations
Get PDFThe purpose of these notes is to give a brief survey of bifurcation theory of Hamiltonian systems with symmetry; they are a slightly extended version of the 5 lectures given by JM on Hamiltonian Systems with Symmetry at the Peyresq Summer School. Attention is focussed on bifurcations near equilibrium solutions and relative equilibria. [Taken from introduction