It is well known that the integrability (solvability) of a differential equation is related to the singularity structure of its solutions in the complex domain-an observation that lies behind the Painleve test. A number of ways of extending this philosophy to discrete equations are explored. First, following the classical work of Julia, Birkhoff and others, a natural interpretation of these equations in the complex domain as difference or delay equations is described and it is noted that arbitrary periodic functions play an analogous role for difference equations to that played by arbitrary constants in the solution of differential equations. These periodic functions can produce spurious branching in solutions and are factored out of the analysis which concentrates on branching from other sources. Second, examples and theorems from the theory of difference equations are presented which show that, module these periodic functions, solutions of a large class of difference equations are meromorphic, regardless of their integrability. It is argued that the integrability of many difference equations is related to the structure of their solutions at infinity in the complex plane and that Nevanlinna theory provides many of the concepts necessary to detect integrability in a large class of equations. A perturbative method is then constructed and used to develop series in z and the derivative of log Gamma(z), where z is the independent variable of the difference equation. This method provides an analogue of the series developed in the Painleve test for differential equations. Finally, the implications of these observations are discussed for two tests which have been studied in the literature regarding the integrability of discrete equations.

We study the local equation of energy for weak solutions of three-dimensional incompressible Navier-Stokes and Euler equations. We define a dissipation term D(u) which stems from an eventual lack of smoothness in the solution u. We give in passing a simple proof of Onsager's conjecture on energy conservation for the three-dimensional Euler equation, slightly weakening the assumption of Constantin et al. We suggest calling weak solutions with non-negative D(u) 'dissipative'.

Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a large-scale neutral mode that must be included in the asymptotic analysis for pattern formation near onset. Near a stationary bifurcation, the usual Ginzburg-Landau equation for the amplitude of the pattern is then coupled to an equation for the large-scale mode. These amplitude equations show that for certain parameters all roll-type solutions are unstable. This new instability differs from the Eckhaus instability in that it is amplitude-driven and is supercritical. Beyond the stability boundary, there exist stable stationary solutions in the form of strongly modulated patterns. The envelope of these modulations is calculated in terms of Jacobi elliptic functions and, away from the onset of modulation, is closely approximated by a sech profile. Numerical simulations indicate that as the modulation becomes more pronounced, the envelope broadens. A number of applications are considered, including convection with fixed-flux boundaries and convection in a magnetic field, resulting in new instabilities for these systems.

We consider travelling wave solutions on a one-dimensional lattice, corresponding to mass particles interacting nonlinearly with their nearest neighbour (the Fermi-Pasta-Ulam model). A constructive method is given, for obtaining all small bounded travelling waves for generic potentials, near the first critical value of the velocity. They all are given by solutions of a finite-dimensional reversible ordinary differential equation. In particular, near (above) the first critical velocity of the waves, we construct the solitary waves (localized waves with the basic state at infinity) whose global existence was proved by Friesecke and Wattis, using a variational approach. In addition, we find other travelling waves such as (a) a superposition of a periodic oscillation with a non-zero uniform stretching or compression between particles, (b) mainly localized waves which tend towards a uniformly stretched or compressed lattice at infinity, (c) heteroclinic solutions connecting a stretched pattern with a compressed one.

In nonlinear dynamics an important distinction exists between uniform bounds on growth rates, as in the definition of hyperbolic sets, and non-uniform bounds as in the theory of Liapunov exponents. In rare cases, for instance in uniquely ergodic systems, it is possible to derive uniform estimates from non-uniform hypotheses. This allowed one of us to show in a previous paper that a strange non-chaotic attractor for a quasiperiodically forced system could not be the graph of a continuous function. This had been a conjecture for some time. In this paper we generalize the uniform convergence of time averages for uniquely egodic systems to a broader range of systems. In particular, we show how conditions on growth rates with respect to all the invariant measures of a system can be used to derive one-sided uniform convergence in both the Birkhoff and the sub-additive ergodic theorems. We apply the latter to show that any strange compact invariant set for a quasiperiodically forced system must support an invariant measure with a non-negative maximal normal Liapunov exponent; in other words, it must contain some 'non-attracting' orbits. This was already known for the few examples of strange non-chaotic attractors that have rigorously been proved to exist. Finally, we generalize our semi-uniform ergodic theorems to arbitrary skew product systems and discuss the application of such extensions to the existence of attracting invariant graphs.

The first definition of Lyapunov exponents (depending on a probability measure) for a one-dimensional cellular automaton was introduced by Shereshevsky in 1991. The existence of an almost everywhere constant value for each of the two exponents (left and right), requires particular conditions for the measure. Shereshevsky establishes an inequality involving these two constants and the metric entropies of both the shift and the cellular automaton. In this paper we first prove that Shereshevsky's two exponents exist for a more suitable class of measures, then, keeping the same conditions, we define new exponents, called average Lyapunov exponents which are smaller than or equal to the former. We obtain two inequalities: the first one is analogous to Shereshevsky's but concerns the average exponents; the second is the Shereshevsky inequality but with more suitable assumptions. These results are illustrated by two non-trivial examples, both proving that average exponents provide a better bound for the entropy, and one showing that the inequalities are strict in general. AMS classification scheme numbers: 37B15, 37A35, 37A25.

Extending the gauge-invariance principle for tau functions of the standard bilinear formalism to the supersymmetric case, we define N = 1 supersymmetric Hirota operators. Using them, we bilinearize SUSY KdV-type equations (KdV, Sawada-Kotera-Ramani). The supersoliton solutions and extension to SUSY sine-Gordon are also discussed. It is shown that the Lax-integrable SUSY KdV of the Mathieu equation does not possess an N-supersoliton solution for N greater than or equal to 3 for arbitrary parameters. The N-supersoliton solution only exists for a particular choice of parameters. AMS classification scheme numbers: 37K10, 35Q51.

We consider the equation x " + mu x(+) - vx(-) = f(x) + g(x) + e(t) where x(+) = max{x, 0}; x(-) = max{-x, 0}, in a situation of resonance for the period 2 pi, i.e. when 1/root mu +1 root upsilon = 2/n for some integer n. We assume that e is 2 pi-periodic, that f has limits f(+/-infinity) at +/-infinity, and that the function g has a sublinear primitive. Denoting by phi a solution of the homogeneous equation x " + mu x(+) - vx(-) = 0, we show that the behaviour of the solutions of the full nonlinear equation depends crucially on whether the function Phi(theta) = n/pi [f(+infinity)/mu - f(-infinity)/upsilon] +1/2 pi integral(0)(2 pi) e(t)phi(t+theta) dt is of constant sign or not. In particular, existence results for 2 pi-periodic and for subharmonic solutions, based on the function Phi, are given.

We study numerically the Cauchy problem for equivariant wave maps from 3 + 1 Minkowski spacetime into the 3-sphere. On the basis of numerical evidence combined with stability analysis of self-similar solutions we formulate two conjectures. The first conjecture states that singularities which are produced in the evolution of sufficiently large initial data are approached in a universal manner given by the profile of a stable self-similar solution. The second conjecture states that the codimension-one stable manifold of a self-similar solution with exactly one instability determines the threshold of singularity formation for a large class of initial data. Our results can be considered as a toy-model for some aspects of the critical behaviour in the formation of black holes. AMS classification scheme numbers: 35L67, 35L70, 35Q75.

The theory of normally hyperbolic invariant manifolds (Fenichel theory) can be used to define strict chaotic synchronization in terms of synchronization manifolds, and treat many ideas found in the physics and engineering literature analytically. In the first part of this work we introduce a modification of Fenichel theory which applies to chaotic synchronization and discuss the Lyapunov-exponent-like quantities used to determine the transverse stability of synchronization manifolds. The second part deals with the different methods for detecting synchrony: symmetry considerations, geometric singular perturbation theory and, in the case of uniformly asymptotically stable extensions, graph transforms. We also consider the case for which an extension of a system is only locally uniformly asymptotically stable and show that in such cases n : 1 synchrony occurs.

We show that in the neighbourhood of the tripling bifurcation of a periodic orbit of a Hamiltonian flow or of a fixed point of an area-preserving map, there is generically a bifurcation that creates a 'twistless' torus. At this bifurcation, the twist, which is the derivative of the rotation number with respect to the action, vanishes. The twistless torus moves outward after it is created and eventually collides with the saddle-centre bifurcation that creates the period-three orbits. The existence of the twistless bifurcation is responsible for the breakdown of the non-degeneracy condition required in the proof of the KAM theorem for flows or the Moser twist theorem for maps. When the twistless torus has a rational rotation number, there are typically reconnection bifurcations of periodic orbits with that rotation number.

The probabilities for gaps in the eigenvalue spectrum of finite N x N random unitary ensembles on the unit circle with a singular weight, and the related Hermitian ensembles on the line with Cauchy weight, are found exactly. The finite cases for exclusion from single and double intervals are given in terms of second-order second-degree ordinary differential equations (ODEs) which are related to certain Painleve-VI transcendents. The scaled cases in the thermodynamic limit are again second degree and second order, this time related to Painleve-V transcendents. Using transformations relating the second-degree ODE and transcendent we prove an identity for the scaled bulk limit which leads to a simple expression for the spacing probability density function. We also relate all the variables appearing in the Fredholm determinant formalism to particular Painleve transcendents, in a simple and transparent way, and exhibit their scaling behaviour. AMS classification scheme numbers: 15A52, 34A34, 34M55, 33C45.

A linear stability analysis of metallic nanowires is performed in the free-electron model using quantum chaos techniques. It is found that the classical instability of a long wire under surface tension can be completely suppressed by electronic shell effects, leading to stable cylindrical configurations whose electrical conductance is a magic number 1, 3, 5, 6,... times the quantum of conductance. Our results are quantitatively consistent with recent experiments with alkali metal nanowires. AMS classification scheme numbers: 76E17, 81Q50, 82D35.

We consider skew-product maps related to dynamics of semigroups generated by rational maps on the Riemann sphere. The entropy of these maps will be given and we will see there exists the unique maximal entropy measure. We will also show the uniqueness of the self-similar measure. We will estimate the Hausdorff dimension of the Julia sets of semigroups. AMS classification scheme numbers: 30D05. 58F23.

We extend previous results obtained by Rosa (1998 Nonlinear Anal. 32 71-85) on the existence of the global attractor for the two-dimensional Navier-Stokes equations on some unbounded domains. We show that if the forcing term is in the natural space H, then the global attractor is compact not only in the L-2 norm but also in the H-1 norm, and it attracts all bounded sets in H in the metric of V. The proof is based on the concept of asymptotic compactness and the use of the enstrophy equation. As compared with the work of Rosa, which proved the compactness and the attraction in the L-2 norm, the new difficulty comes from the fact that the nonlinear term of the Navier-Stokes equations does not disappear from the enstrophy equation, while it does disappear in the energy equation due to its antisymmetry property. AMS classification scheme numbers: 76D05, 34D45, 35B41.

Recently, a number of approaches have been developed to connect the microscopic dynamics of particle systems to the macroscopic properties of systems in non-equilibrium stationary states, via the theory of dynamical systems. In this way a direct connection between dynamics and irreversible thermodynamics has been claimed to have been found. However, the main quantity used in these studies is a (coarse-grained) Gibbs entropy, which to us does not seem suitable, in its present form, to characterize non-equilibrium states. Various simplified models have also been devised to give explicit examples of how the coarse-grained approach may succeed in giving a full description of the irreversible thermodynamics. We analyse some of these models and point out a number of difficulties which, in our opinion, need to be overcome in order to establish a physically relevant connection between these models and irreversible thermodynamics. AMS classification scheme numbers: 82C05, 80A20, 70F25.

We discuss the existence of large isolated (non-unit) eigenvalues of the Perron-Frobenius operator for expanding interval maps. Corresponding to these eigenvalues (or 'resonances') are distributions which approach the invariant density (or equilibrium distribution) at a rate slower than that prescribed by the minimal expansion rate. We consider the transitional behaviour of the eigenfunctions as the eigenvalues cross this 'minimal expansion rate' threshold, and suggest dynamical implications of the existence and form of these eigenfunctions. A systematic means of constructing maps which possess such isolated eigenvalues is presented. AMS classification scheme numbers: 37A30(primary), 37E05, 37D20, 47A10, 47A15 (secondary).

In this paper an upper estimate of the number of limit cycles of the Abel equation (x)over dot = v(n, t), x is an element of R, is an element of S-1 is given. Here v is a polynomial in x with the higher coefficient one and periodic in t with period one. The bound depends on the degree n of the polynomial and the magnitude of its coefficients. In the second part we give an explicit upper estimate of the number of zeros of a holomorphic function in a compact subset of its domain through the growth rate of the function and some geometric constant that is expressed here by means of the Poincare metric. This improves the estimate given in Ilyashenko and Yakovenko (1996 J. Differ Equ. 126 87- 105). AMS classification scheme numbers: 34C07, 32A10, 37C10.

We study the behaviour of the standard map critical function in a neighbourhood of a fixed resonance, that is the scaling, law at the fixed resonance. We prove that for the fundamental resonance the scaling law is linear. We show numerical evidence that for the other resonances p/q, q greater than or equal to 2, p not equal 0 and p and q relatively prime, the scaling law follows a power-law with exponent 1/q. AMS classification scheme numbers: 37C55, 37E40, 70K43, 34C28.