Mathematical theories describing chaotic behaviour in physical systems are introduced by developing and reviewing applications to optical fibres. A theory is presented for laser light propagating in a loop formed by an optical fibre and an optical coupler. As the light traverses the fibre it suffers an attenuation and is subjected to a phase shift which will have a component proportional to the light intensity via the nonlinear optics Kerr effect. At each pass through the coupler, an extra fraction of laser light is coupled into the loop. The mathematical formulation leads to a two-dimensional map having a clear physical and geometrical interpretation. The complete solution is given in the linear regime and the onset of nonlinear behaviour is investigated as the laser power is increased. A variety of transitions is obtained including period doubling and iteration onto a strange attractor.

“Steady state” periodic solutions are sought to the forced Duffing equation. The solutions are expressed as formal Fourier series, giving rise to an infinite system of non-linear algebraic equations for the Fourier coefficients. This system is then solved using perturbation series in the amplitude of the forcing term. Solution profiles of high accuracy and phase-plane orbits are presented. The existence of limiting values of the forcing amplitude is discussed, and points of non-linear resonance are identified.

It is shown that barrier functions applied to the dual linear program can be modified to give multiplier estimates that converge to the solution of the primal problem. Newton's method is considered for implementing this approach and numerical results presented. It has been shown that there is a connection between these methods and Karmarkar's algorithm, but for the class of problems considered further improvements are still required before those methods become competitive with active set methods.

It is well known that n-process, n-commodity (or square), models of productive single-product industries have positive solutions to their price and quantity systems if the rates of profit and growth lie in appropriate non-negative intervals. On the other hand, negative prices and quantities can occur in formal solutions of models of square, productive, multiple-product industries even when the rates of profit and growth are less than their respective maximum positive values. It is shown in this paper that these differences can be attributed to the presence in joint production of dominance, in either row or column versions. Results on positive solutions to the price (respectively, quantity) system are derived in terms of the absence of column (respectively, row) dominance of the net output matrix. As the concepts of row and column dominance are defined in terms of linear inequalities, the basic mathematical results to be applied are theorems of the alternative.

The spectral function , where are the eigenvalues of the two-dimensional Laplacian, is studied for a variety of domains. The dependence of θ(t) on the connectivity of a domain and the impedance boundary conditions is analysed. Particular attention is given to a doubly-connected region together with the impedance boundary conditions on its boundaries.

Error estimates are derived for a finite element analysis of plane steady subsonic flows described by the full potential equation. The analysis is based on the use of the theory of variational inequalities to accomodate the subsonic flow constraint and leads to a suboptimal estimate relative to that obtained for linear potential flow. We then consider an alternative dual formulation of the problem and obtain an optimal estimate subject to reasonable regularity assumptions.

Short-crested waves are defined as propagating surface gravity waves which are doubly-periodic in the horizontal plane. Linearly, the short-crested wave system we consider occurs when two progressive wavetrains of equal amplitude and frequency are propagating at an angle to each other.

Solutions are calculated via a computer-generated perturbation expansion in wave steepness. Harmonic resonance affects the solutions but Padé approximants can be used to estimate wave properties such as maximum wave steepness, frequency, kinetic energy and potential energy.

The force exerted by waves being reflected by a seawall is also calculated. Our results for the maximum depth-integrated onshore wave force in the standing wave limit are compared with experiment. The maximum force exerted on a seawail occurs for a steep wave in shallow water incident at an oblique angle. Results are given for this maximum force.