The Bonhoeffer Van der Pol system is a planar autonomous nonlinear system of differential equations which has been invoked as a qualitative model of physiological states in a nerve membrane. It contains three independent parameters and previous work has only studied a small portion of the parameter space, that part which is thought to be of physiological relevance. Here we give a complete study of the full parameter space, using both theoretical results and numerical solutions.

By using the spectral Galerkin method, we prove a result on the global existence in time of strong solutions for a system of equations of magnetohydrodynamic type. Several estimates for the solution and their approximations are given. These estimates can be used in the derivation of error bounds for the approximate solutions.

The change of impedance per unit length in a single or double conductor line situated parallel to an infinitely long two-layered metallic circular cylinder is found (within the quasistatic approximation) in the form of an infinite series. The cylinder consists of an inner core and an outer annulus. The properties of the inner core are assumed to be constant. The relative magnetic permeability, μ(r) = rα and the conductivity, σ(r) = σ(0)rκ, of the outer annulus vary with respect to the radial coordinate, r, and α and κ are arbitrary real numbers. Numerical results are presented in the form of figures and tables.

For a predator-prey model with time-delay due to gestation, criteria are obtained for persistence and global attractivity. The global attractivity criteria apply only to models with a decreasing prey isocline.

In a recent paper, Christie and Gopalsamy [2] used Melnikov's method to establish a sufficient condition for the existence of chaotic behaviour, in the sense of Smale, in a particular time-periodically perturbed planar autonomous system of ordinary differential equations. They then concluded with an application to the dynamics of a one-dimensional anharmonic oscillator. In this paper, the same system is considered and a condition for the existence of subharmonic orbits in the perturbed system is deduced, using the subharmonic Melnikov theory. Finally, an application is given to the dynamical behaviour of the one-dimensional anharmonic oscillator system.

This paper examines the role of import tariffs and consumption taxes when a product is supplied to a domestic market by a foreign monopoly via a subsidiary. It is assumed that there is no competition in the domestic market from internal suppliers. The home country is able to levy a profits tax on the subsidiary. The objective of our analysis is to determine the mix of tariff and consumption tax which simultaneously maximizes national welfare. We show that national welfare does not have an internal maximum, but attains its maximum on a boundary of the consumption tax–tariff parameter space. Furthermore, the optimal value of national welfare increases as the tariff decreases and the consumption tax increases. The results obtained generalize the results of an earlier paper in which national welfare was maximized with respect to either a tariff or consumption tax, but not both.

The authors consider the higher-order nonlinear neutral delay difference equation

and obtain results on the asymptotic behavior of solutions when (pn) is allowed to oscillate about the bifurcation value –1. We also consider the case where the sequence {pn} has arbitrarily large zeros. Examples illustrating the results are included, and suggestions for further research are indicated.

This note introduces quasi-local-balance for discrete-time Markov chains with absorbing states. From quasi-local-balance product-form quasi-stationary distributions are derived by analogy with product-form stationary distributions for Markov chains that satisfy local balance.

Applying Ekeland's variational principle in this paper, we obtain a maximum principle for optimal control for a class of two-point boundary value controlled systems. The control domain need not be convex. For a special case, that is the so called LQ-type problem, we obtain the optimal control in the closed loop form and a corresponding Riccati type differential equation.

Selective withdrawal from a stratified stream is considered. The average density of the withdrawn fluid and the flow pattern are found, within the limitations on the densimetric Froude number and the withdrawal rate specified in this paper, to depend on the strength and location of the sink, and very little on any slight variation in the velocity distribution far upstream and the densimetric Froude number. The upstream density distribution is assumed linear, but many other density distributions can be similarly treated.

The paper is concerned with formation of singularities in a density stratified fluid subject to a monochromatic point source of frequency σ. The frequency of the source is assumed to be such that the steady-oscillation equation is hyperbolic in the neighbourhood of the source and degenerates at a critical level. We obtain asymptotic formulae demonstrating how the solution diverges as t → ∞ on the characteristic surface emanating from the source. It is shown that, at points of the surface that belong to the critical level, the solution behaves as t⅔ exp {i(σt + π/2)} as t → ∞, whereas its large time behaviour at the other points of the surface is given by t½ exp {i(σt + π/2 ± π/4)}.

Finding critical phenomena in two-dimensional combustion is normally done numerically. By using a centre-manifold reduction, we can find a reduced equation in one dimension. Once we have found the reduced equation, it is simpler to find critical phenomena. We consider two different problems. One is spontaneous ignition. We compare our results with known critical parameters to give some validity to our reduction technique. We also look at a combustion model with three equilibrium states. For this model, the possible transitions can occur as travelling waves between the unstable to either of the stable equilibrium or from one stable to the other stable state. For the latter transition, the direction of the transition tells us whether we have an extinction or ignition wave. We find the critical parameters when the direction of the wave changes.

The allometric hypothesis which relates the shape (y) of biological organs to the size of the plant or animal (x), as a function of the relative growth rates, is ubiquitous in biology. This concept has been especially useful in studies of carcass composition of farm animals, and is the basis for the definition of maintenance requirements in animal nutrition.

When the size variable is random the differential equation describing the relative growth rates of organs becomes a stochastic differential equation, with a solution different from that of the deterministic equation normally used to describe allometry. This is important in studies of carcass composition where animals are slaughtered in different sizes and ages, introducing variance between animals into the size variable.

This paper derives an equation that relates values of the shape variable to the expected values of the size variable at any point. This is the most easily interpreted relationship in many applications of the allometric hypothesis such as the study of the development of carcass composition in domestic animals by serial slaughter. The change in the estimates of the coefficients of the allometric equation found through the usual deterministc equation is demonstrated under additive and multiplicative errors. The inclusion of a factor based on the reciprocal of the size variable to the usual log - log regression equation is shown to produce unbiased estimates of the parameters when the errors can be assumed to be multiplicative.

The consequences of stochastic size variables in the study of carcass composition are discussed.

The resolution of many problems in probability depends on being able to provide sufficiently good upper or lower bounds to certain moments of distributions. A striking example from the literature of a result that can offer such bounds was given by Pó1ya over sixty years ago as the following theorem (see [7, Vol. II, p. 144] and [7, Vol. I, p. 94]).

In this paper we establish conditions which ensure the existence of self-excited oscillations in complex dynamical systems with nondifferentiable nonlinearities, by considering those types of systems which can be viewed as an interconnection of several simpler subsystems. We find that the nonlinear terms of the system in which we are interested do not need to satisfy the Lipschitz condition.

A class of optimal control models which involve different weightings in the integrand of the objective function is considered. The motivation for considering this class of problems is that this type of objective function is used to account for eccentric movement in biomechanical models. The computation of these optimal control problems using control parametrization directly is difficult, firstly because of ill-conditioning, and secondly because the objective function is not differentiable. A method for smoothing the integrand is presented with convergence results. An example is computed which shows favourable computational improvements.

It is known that many optimization problems can be reformulated as composite optimization problems. In this paper error analyses are provided for two kinds of smoothing approximation methods of a unconstrained composite nondifferentiable optimization problem. Computational results are presented for nondifferentiable optimization problems by using these smoothing approximation methods. Comparisons are made among these methods.

Steady two-dimensional flows in a domain bounded below by an infinite horizontal wall and above by a semi-infinite horizontal wall, a vertical wall and a free surface are considered. The fluid is assumed to be inviscid and incompressible, and gravity is taken into account. The problem is solved numerically by series truncation. It is shown that for a given length of the vertical wall, there are two families of solutions. One family is characterized by a continuous slope at the separation point and a limiting configuration with a stagnation point and a 120° angle corner at the separation point. The other family is characterized by a stagnation point and a 90° angle corner at the separation point. Flows under a sluice gate with and without a rigid lid approximation upstream are also considered.

In this paper we present an adaptive boundary-element method for a transmission prob-lem for the Laplacian in a two-dimensional Lipschitz domain. We are concerned with an equivalent system of boundary-integral equations of the first kind (on the transmission boundary) involving weakly-singular, singular and hypersingular integral operators. For the h-version boundary-element (Galerkin) discretization we derive an a posteriori error estimate which guarantees a given bound for the error in the energy norm (up to a multiplicative constant). Then, following Eriksson and Johnson this yields an adaptive algorithm steering the mesh refinement. Numerical examples confirm that our adaptive algorithms yield automatically good triangulations and are efficient.

In the present paper questions related to stability and boundedness with respect to manifolds of solutions of impulsive differential-difference equations are considered. The investigations are carried out by means of piecewise-continuous functions which are analogues of the classical Lyapunov's functions. By means of a vectorial comparison equation and differential inequalities for piecewise-continuous functions, theorems are proved on stability and boundedness with respect to manifolds of solutions of impulsive differential-difference equations with impulse effect at fixed moments.