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Analysis of a nonlinear dynamics model of the saccadic system
Get PDFModels of the mechanisms of saccadic eye movements are typically described in terms of the block diagrams used in control theory. Recently, a nonlinear dynamics model of the saccadic system was developed. The model comprises a symmetric piecewise-smooth system of six first-order autonomous ordinary differential equations, which were obtained by combining parts of the existing control models with data from experimental observations of saccadic dynamics. A preliminary numerical investigation of the model revealed that in addition to generating normal saccades, it could also simulate inaccurate saccades, and an oscillatory instability known as congenital nystagmus (CN). By varying the parameters of the model, several types of CN oscillations could be produced, including jerk, bilateral jerk and pendular nystagmus.
The aim of this study was to investigate the bifurcations and attractors of the nonlinear dynamics model, in order to obtain a classification of the simulated oculomotor behaviours. The application of standard local and global stability analysis techniques, together with numerical work, revealed that the equations have a rich bifurcation structure. In addi- tion to Hopf, homoclinic and saddlenode bifurcations organised by a Takens-Bogdanov point, the equations can undergo nonsmooth pitchfork bifurcations and nonsmooth glu- ing bifurcations. These nonsmooth bifurcations were observed to result from simultaneous transcritical and homoclinic bifurcations in a pair of related smooth systems. Evidence was also found for the existence of Hopf-initiated canards, and for a global bifurcation involving the catastrophic destruction of a symmetry-invariant limit cycle. Unlike the pitchfork and gluing bifurcations, this bifurcation could not be explained in terms of the related smooth systems.
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 equa- tions are a slow-fast system. The slow and fast phases of the simulated oscillations were attributed to the geometry of an underlying slow manifold. This provides an alternative explanation for the shape of the jerk oscillation, which contrasts with the prevalent control model view that CN is caused by structural abnormalities. The simulated bilateral jerk and pendular waveforms were attributed to a symmetry invariant limit cycle produced by the gluing of the asymmetric cycles.
The bifurcation structure of the model suggests the possibility of moving between the differ- ent simulated behaviours by varying the parameters of the model. This was in agreement with experimental evidence showing that sub jects can exhibit several different types of behaviour in a single recording period. In addition, the bifurcation analysis places restric- tions on which kinds of behaviour are likely to be associated with each other in parameter space. On the basis of these restrictions, several experiments were suggested to assess the validity of the model as a predictor of saccadic behaviour. In particular, it was proposed that reducing the level of attention of a sub ject in a controlled way could induce a change from a jerk to a pendular oscillation
Flow phase diagrams for concentration-coupled shear banding
No full textAfter surveying the experimental evidence for concentration coupling in the shear banding of wormlike micellar surfactant systems, we present flow phase diagrams spanned by shear stress (or strain rate ) and concentration, calculated within the two-fluid, non-local Johnson-Segalman (d-JS- ) model. We also give results for the macroscopic flow curves for a range of (average) concentrations . For any concentration that is high enough to give shear banding, the flow curve shows the usual non-analytic kink at the onset of banding, followed by a coexistence "plateau" that slopes upwards, \drm \Sigma/ \drm \bar{\dot{\gamma}}>0. As the concentration is reduced, the width of the coexistence regime diminishes and eventually terminates at a non-equilibrium critical point . We outline the way in which the flow phase diagram can be reconstructed from a family of such flow curves, , measured for several different values of . This reconstruction could be used to check new measurements of concentration differences between the coexisting bands. Our d-JS- model contains two different spatial gradient terms that describe the interface between the shear bands. The first is in the viscoelastic constitutive equation, with a characteristic (mesh) length l. The second is in the (generalised) Cahn-Hilliard equation, with the characteristic length for equilibrium concentration-fluctuations. We show that the phase diagrams (and so also the flow curves) depend on the ratio , with loss of unique state selection at r=0. We also give results for the full shear-banded profiles, and study the divergence of the interfacial width (relative to l and ) at the critical point
Asymptotic matching constraints for a boundary-layer flow of a power-law fluid
Get PDFWe reconsider the three-dimensional boundary-layer flow of a power-law (Ostwald–de Waele) rheology fluid, driven by the rotation of an infinite rotating plane in an otherwise stationary system. Here we address the problem for both shear-thinning and shear-thickening fluids and show that there are some fundamental issues regarding the application of power-law models in a boundary-layer context that have not been mentioned in previous discussions. For shear-thickening fluids, the leading-order boundary-layer equations are shown to have no suitable decaying behaviour in the far field, and the only solutions that exist are necessarily non-differentiable at a critical location and of ‘finite thickness’. Higher-order effects are shown to regularize the singularity at the critical location. In the shear-thinning case, the boundary-layer solutions are shown to possess algebraic decay to a free-stream flow. This case is known from the existing literature; however here we shall emphasize the complexity of applying such solutions to a global flow, describing why they are in general inappropriate in a traditional boundary-layer context. Furthermore, previously noted difficulties for fluids that are highly shear thinning are also shown to be associated with the imposition of incorrect assumptions regarding the nature of the far-field flow. Based on Newtonian results, we anticipate the presence of non-uniqueness and through accurate numerical solution of the leading-order boundary-layer equations we locate several such solutions
On the embedding problem for infinitely divisible distributions on certain Lie groups with toral center
No full textScaling of the threshold of pipe flow turbulence
No full textWe report the results of an experimental investigation of the transition to turbulence in a pipe over approximately an order of magnitude range in the Reynolds number Re. A novel scaling law is uncovered using a systematic experimental procedure which permits contact to be made with modern theoretical thinking. The principal result we uncover is a scaling law which indicates that the amplitude of perturbation required to cause transition scales as O(Re-1)
Coxeter Matroids
No full textMatroids appear in diverse areas of mathematics, from combinatorics to algebraic topology and geometry. This largely self-contained work provides an intuitive and interdisciplinary treatment of Coxeter matroids, a new and beautiful generalization of matroids which is based on a finite Coxeter group.
Key topics and features:
* Systematic, clearly written exposition with ample references to current research
* Matroids are examined in terms of symmetric and finite reflection groups
* Finite reflection groups and Coxeter groups are developed from scratch
* The Gelfand-Serganova Theorem is presented, allowing for a geometric interpretation of matroids and Coxeter matroids as convex polytopes with certain symmetry properties
* Matroid representations and combinatorial flag varieties are studied in the final chapter
* Many exercises throughout
* Excellent bibliography and index
Accessible to graduate students and research mathematicians alike, Coxeter Matroids can be used as an introductory survey, a graduate course text, or a reference volume
On the diffusion coefficient: The Einstein relation and beyond
No full textWe present a detailed derivation of the closed-form expression for the diffusion coefficient that was initially obtained by Einstein.[4] The present derivation does not make use of a fictitious force as did the original Einstein derivation, but instead concentrates directly on establishing a dynamic equilibrium between the forces of pressure and friction acting on a Brownian particle. This approach makes it easier to understand the true essence of the argument, and thus makes it simpler to apply the argument in a more general case or setting. We demonstrate this by deriving the equation of motion of a Brownian particle that is under the influence of an external force in the fluid with a non-constant temperature. This equation extends the well-known Smoluchowski approximation[24] to the case of non-constant temperature, and offers new insights into the Ludwig–Soret and Enskog–Chapman effects (providing also a scholar example explaining the need for a stochastic integral). The key point in the derivation is reached by applying the Einstein dynamic equilibrium argument together with the conservation of the number of particles law. We show that this approach leads directly to the Kolmogorov forward equation whenever the setting is Markovian. The same method can also be applied in the case of interacting Brownian particles satisfying the van der Waals equation. In this setting we first demonstrate that the presence of short-range repulsive forces between Brownian particles tends to increase the diffusion coefficient, and the presence of long-range attractive forces between Brownian particles tends to decrease it. The method of derivation then leads to a nonlinear partial differential equation which in the case of weak interaction reduces to the Fokker–Planck equation. One of the main aims of the present article is to demonstrate that the Einstein argument leads to a truly dynamical theory of diffusion
A study of stochastic differential equations with non-Lipschitzian coefficients
No full textWe study a class of stochastic differential equations with non-Lipschitz coefficients. A unique strong solution is obtained and the non confluence of the solutions of stochastic differential equations is proved. The dependence with respect to the initial values is investigated. To obtain a continuous version of solutions, the modulus of continuity of coefficients is assumed to be less than |x-y| log MediaObjects/s00440-004-0398-zflb1.gif Finally a large deviation principle of
Freidlin-Wentzell type is also established in the paper
The unsteady Kàrmàn problem for a dilute particle suspension
Get PDFWe consider the unsteady three-dimensional Kármán flow induced by the impulsive rotation of an infinite rotating plane immersed in an incompressible viscous fluid with a dilute suspension of small solid monodisperse spherical particles. The flow is described in terms of a ‘dusty gas’ model, which treats the discrete phase (particles) and the continuous phase (fluid) as two continua occupying the same space and interacting through a Stokes drag mechanism. The model is extended to allow for a local gravitational acceleration in a direction parallel to the axis of rotation, and is valid for cases in which gravity acts either in the same direction as or in the opposite direction to the Ekman axial flow induced by the rotation of the plane.
Analysis based on the theory of characteristics shows that the role of gravity is crucial to the treatment of the discrete-phase equations, particularly in regard to the appropriate boundary conditions to be applied at the solid surface. Other notable features include the presence of an essential singularity in the solution when gravity is absent; indeed this phenomenon may help to explain some of the difficulties encountered in previous studies of this type. If the gravitational force is directed away from the rotating surface, a number of other interesting features arise, including the development of discontinuities in the particle distribution profiles, with corresponding particle-free regions contained between the interface and the rotating boundary. These ‘shock’ features can be associated with a critical axial location in the boundary layer at which a balance is achieved between Ekman suction induced by the rotating boundary and the influence of gravitational effects acting to move particles away from the boundary