Table of contents

This year's cover illustration shows an unexpected singular structure occurring in a flow seen in an ordinary household kitchen sink. When a vertical jet of liquid impinges on a flat surface it creates a ring discontinuity, the circular hydraulic jump, at a well-defined distance from the jet, where the depth of the fluid layer changes by an order of magnitude. By using a liquid more viscous than water, we have noticed that the circular hydraulic jump may spontaneously deform to a polygonal structure breaking the axial symmetry. The corners can be very sharp and carry a large radial flux while the edges (sides) generate resistance on the stream. We describe this experiment in detail, and present a heuristic explanation of the rather aesthetically pleasing states. The explanation is admittedly not systematic, but illuminates the role of the internal eddy structure in the flow, and qualitatively agrees with the observed parameter dependence.

Pecora and Carroll presented a notion of synchronization where an (n - 1)-dimensional nonautonomous system is constructed from a given n-dimensional dynamical system by imposing the evolution of one coordinate. They noticed that the resulting dynamics may be contracting even if the original dynamics are not. It is easy to construct flows or maps such that no coordinate has synchronizing properties, but this cannot be done in an open set of linear maps or flows in , . In this paper we give examples of real analytic homeomorphisms of such that the nonsynchronizability is stable in the sense that in a full neighbourhood of the given map, no homeomorphism is synchronizable.

Using thermodynamical formalism, we set up the perturbative approach for the Hausdorff dimension and the dimension spectrum of the Julia set associated with polynomials close to monomials. We extend previous calculations to a more general case where the perturbation is not constant.

In a recent paper Sherratt and Marchant (1996 IMA J. Appl. Math. 56 289-302) have made numerical studies of initial value problems for the cubic Fisher equation. It was observed that for initial data with sufficiently slow decay rate at large x (dimensionless distance), the long-time evolution involved the development of an accelerating and stretching wavefront, rather than the more usual permanent form, constant velocity reaction-diffusion wavefront. They also presented a heuristic argument to obtained an expression for the propagation speed of this wavefront. In this paper we address the same initial value problem for the generalized Fisher equation of order . Using matched asymptotic expansions we are able to obtain the complete structure of the solution to the initial value problem for large time. This confirms the computations of Sherratt and Marchant for m > 1, and gives complete information about the structure and propagation speed of the evolving, accelerating, phase wave. In addition, we also find that phase waves can propagate in the case m = 1, which is contrary to the suggestions of Sherratt and Marchant. For m = 1, the asymptotics are exponential in t (dimensionless time), whilst for m > 1 the asymptotics are algebraic in t.

In this paper, we study 1-homoclinic loop bifurcations on a non-orientable 2-manifold: the Möbius band. The techniques for studying bifurcating dynamics of the 1-homoclinic loop on this manifold are similar to those for a figure-eight loop in the plane. We adapt the techniques revealed in a paper by Jebrane and Mourtada (1994 Cyclicité finie des lacets doubles non triviaux 7 1349-65) covering the subject: we are able, studying the 2-return map where it exists, to give an explicit bound for the cyclicity of the 1-homoclinic loop for all arbitrary finite codimensions. The key element is a blow-up. A simple corollary is to prove the completeness of the bifurcation diagram given by Chow et al (1990 Homoclinic bifurcation at resonant eigenvalues, J. Dyn. Diff. Equ. 2 177-244).

Using a special form of Ulam's method, we estimate the measure-theoretic entropy of a triple , where M is a smooth manifold, T is a uniformly hyperbolic map, and is the unique physical measure of T. With a few additional calculations, we also obtain numerical estimates of (i) the physical measure , (ii) the Lyapunov exponents of T with respect to , (iii) the rate of decay of correlations for with respect to test functions, and (iv) the rate of escape (for repellors). Four main situations are considered: T is everywhere expanding, T is everywhere hyperbolic (Anosov), T is hyperbolic on an attracting invariant set (axiom A attractor), and T is hyperbolic on a non-attracting invariant set (axiom A non-attractor/repellor).

Solitary waves in conservative and near-conservative systems may become unstable due to a resonance of two internal oscillation modes. We study the parametrically driven, damped nonlinear Schrödinger equation, a prototype system exhibiting this oscillatory instability. An asymptotic multi-scale expansion is used to derive a reduced amplitude equation describing the nonlinear stage of the instability and supercritical dynamics of the soliton in the weakly dissipative case. We also derive the amplitude equation in the strongly dissipative case, when the bifurcation is of the Hopf type. The analysis of the reduced equations shows that in the undamped case the temporally periodic spatially localized structures are suppressed by the nonlinearity-induced radiation. In this case the unstable stationary soliton evolves either into a slowly decaying long-lived breather, or into a radiating soliton whose amplitude grows without bound. However, adding a small damping is sufficient to bring about a stably oscillating soliton of finite amplitude.

Given a map and a function , a `Fredholm determinant' can be defined as a formal power series . The coefficients are related to the periodic points of . Assume that is a hyperbolic diffeomorphism of class , and belongs to . Then the Fredholm determinant is analytic in the disc of radius , where is a hyperbolicity index of (roughly speaking, is -contracting in one direction and -expanding in the other direction). In the case, the Fredholm determinant is an entire function.