Table of contents

Two canonical pattern forming systems, the Rayleigh-Benard convection and the Turing mechanism for biological pattern formation, are compared. The similarity and fundamental differences in the mathematical structure of the two systems are addressed, with special emphasis on how the linear onset of patterns is affected by the finite size and the boundary conditions. Our analysis is facilitated by continuously varying the boundary condition, from one that admits simple algebraic solution of the problem but is unrealistic to another which is physically realizable. Our investigation shows that the size dependence of the convection problem can be considered generic, in the sense that for the majority of boundary conditions the same trend is to be observed, while for the corresponding Turing mechanism one will rely crucially on the assumed boundary conditions to ensure that a particular sequence of patterns be picked up as the system grows in size. This suggests that, although different systems might exhibit similar pattern forming features, it is still possible to distinguish them by characteristics which are specific to the individual models.

A class of mappings of the 2-torus T² onto itself, arising in a model of modulated diffusion is considered. These transformations can be written as a skew-product of the endomorphism x'=2x mod (0,1( on the 1-torus T¹:=(0,1( and a translation on T¹. Ergodicity and mixing with respect to the Haar measure on T² are rigorously proved for any choice of the parameters.

A parametrically forced nonlinear Schrodinger equation is considered. When the forcing is time independent the large time asymptotic solution is found and is shown to correspond to the bound states of an associated linear Schrodinger equation. Special solutions are calculated and the relationship of this problem to other well known nonlinear evolution equations is discussed.

We investigate the statistical distribution of the zeros of Dirichlet L-functions both analytically and numerically. Using the Hardy-Littlewood conjecture about the distribution of primes we show that the two-point correlation function of these zeros coincides with that for eigenvalues of the Gaussian unitary ensemble of random matrices, and that the distributions of zeros of different L-functions are statistically independent. Applications of these results to Epstein's zeta functions are briefly discussed.

The purpose of this paper is to relate the non-existence of polynomial integrals for a Hamiltonian system to the breakdown phenomenon of smooth solutions in quasi-linear equations. Using this relation it is shown that for the classical Hamiltonian system with 1.5 degrees of freedom there are no non-trivial third power integrals of motion. The main tool used in the proof is the Lax analysis on formation of singularities in quasi-linear equations. Some results and perspectives for the case of higher degrees are discussed.

We investigate the existence of a global semiflow for the complex Ginzburg-Landau equation on the space of bounded functions in infinite domain. This semiflow is proven to exist in dimension 1 and 2 for any parameter values of the standard cubic Ginzburg-Landau equation. In dimension 3 we need some restrictions on the parameters but cover nevertheless some part of the Benjamin-Feir unstable domain.

We examine the change in behaviour of the solutions of a simple one degree of freedom, periodically forced, impact oscillator following a grazing bifurcation in which an impact of zero velocity occurs following a change in one of the parameters of the system. It is shown that such a bifurcation leads to intermittent chaotic behaviour with low velocity impacts followed by an irregular sequence of high velocity impacts. We also show that there is a natural, discontinuous one-dimensional map associated with this relating one low velocity impact to the next and the properties of this map are analysed. We also construct the bifurcation diagram of the change in behaviour and show that this contains a series of periodic windows, with the period of the solutions increasing monotonically by one in each successive window as the bifurcation point is approached. By restricting our attention to the resonant case where the forcing frequency is twice the natural frequency of the oscillator it is possible to make asymptotic estimates of the form of the intermittent chaotic behaviour and these estimates are compared with some numerical calculations.

We propose a new method for studying asymptotic behaviour of complex separatrices of some symplectic maps. Our approach gives an alternative method (as opposed to iterating the maps) for computing some constants to measure the splitting of separatrices. Details are presented for semistandard and cubic standard-like maps.

Let H(x,y) be a polynomial of degree n in two variables and omega a polynomial 1-form P(x,y)dx+Q(x,y)dy of degree m. The integral of the form over a continuous family of closed curves on the level sets (H=t) can be extended to a complete Abelian integral, a multivalued analytic function I omega (t) of a point t in C varying over the set of regular values of the polynomial. We prove that this function may be represented as a linear combination of a certain family of analytic multivalued functions depending only on the polynomial H. The coefficients of this combination are rational functions of the variable t of degrees growing linearly with m=deg omega to + infinity . For a generic Hamiltonian H we obtain an upper estimate for those degrees in the form ^m/_n+ⁿ(6)/₂.

A C² diffeomorphism of the two-sphere, whose periodic points are hyperbolic and infinitely many, can be C¹ approximated by a diffeomorphism with homoclinic points.

The interface between an unstable state and a stable state usually develops a single confined front travelling with constant velocity into the unstable state. Recently, the splitting of such an interface into two fronts propagating with different velocities was observed numerically in a magnetic system. The intermediate state is unstable and grows linearly in time. We first establish rigorously the existence of this phenomenon, called a 'dual front', for a class of structurally unstable one-component models. Then we use this insight to explain dual fronts for a generic two-component reaction-diffusion system, and for the magnetic system.

When -2<or=c<or=1/4, the Julia set J_c of z to z²+c is connected. We construct a family of ellipses E_c which contain J_c when -2<or=c<or=0, and which are contained within J_c when 0<or=c<or=1/4. The ellipse E₀ coincides with J₀, (the unit circle).