A form of the Hopf bifurcation theorem specially suited to systems in block-diagram form has been developed, which allows one to deal with high or infinite order linear elements solely in terms of their transfer functions. The results are proved by the method of harmonic balance, and, for general non-linear systems, lead to criteria for the existence and stability of bifurcated orbits generalizing those derived by various authors for systems of ordinary differential equations. In the particular case of control loops with a single non-linearity, a simple addition to the Nyquist diagram of the loop determines the amplitude and frequency of bifurcated orbits, and whether they occur when the equilibrium is stable or unstable. The analysis is independent of the central manifold theorem, and of Floquet theory.Introduction. In a system of differential equations X = f^X) (for vector X), dependent on a real parameter /t, the stability of an equilibrium point a is determined by the characteristic exponents of the linearization of the system about a. It may be that a is stable for /i < /i 0) but that as fi increases above fi 0 , a pair of exponents cross the imaginary axis, so that a is unstable for /i > /i 0 . Then E. Hopf (2) showed that, under certain conditions, this change is associated with the appearance or disappearance of periodic (non-constant) solutions in the neighbourhood of a. There are two distinct ways in which this association can occur: either the periodic solutions are (orbitally) stable and exist when the equilibrium is unstable, or the periodic solutions are unstable and exist when the equilibrium is stable. We shall call these type I and type II bifurcations respectively. There are examples for which the periodic solutions only exist when ft is exactly equal to fi 0 or for which two or more distinct families of periodic solutions exist in the neighbourhood of a, and we shall not count these in either of the two types. A full discussion of the Hopf bifurcation theorem and its applications appears in (3).In § 1, we apply the method of harmonic balance to a system with a single non-linear element; this leads to a simple addition to the Nyquist diagram that determines the type of a bifurcation, and the amplitude and frequency of the bifurcated orbits, correct to the first order in /*-fi 0 . Even when /i-/i o is not small, this provides a useful approximate graphical method of applying second-order harmonic balance to such systems. The analysis in § 1 is not done rigorously, but illustrates ideas that are important for the work of § 2, where we generalize the system considered, and prove the main theorem giving existence of bifurcated orbits, and determining the type of a bifurcation. We show that the formulae derived give those of § 1 for the particular systems considered there, and also generalize those recently derived by A. B. Poore(7) for systems of ordinary differential equations. In §3 we demonstate local uniqueness and stability

A mathematical methodology for analysing pile-ups of large numbers of dislocations is described. As an example, the pile-up of n identical screw or edge dislocations in a single slip plane under the action of an external force in the direction of a locked dislocation in that plane is considered. As n ! 1 there is a well-known formula for the number density of the dislocations, but this density is singular at the lock and it cannot predict the stress field there or the force on the lock. This poses the interesting analytical and numerical problem of matching a local discrete model near the lock to the continuum model further away. r

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Elastic buckling of a spherical shell, embedded in an elastic material and loaded by a far-field hydrostatic pressure is analysed using the energy method together with a Rayleigh—Ritz trial function. For simplicity, only axisymmetric deformations are considered and inextensional buckling is assumed. The strains within the structure that are pre-critical are assumed to be small for the linear theory to be applicable. An expression is derived relating the pressure load to the buckling mode number, from which the upper-bound critical load can be determined. It is found that the presence of the surrounding elastic medium increases the critical load of the shell and the corresponding buckling mode number. However, the results also show that the strain of the shell at the point of instability may not be small for typical values of material and geometric constants.