Table of contents

This year's cover illustration, reproduced as figure 4, comes from the branch of mathematics called complex dynamics. It shows a subset of the parameter space of a certain family of maps.

Using Sinai - Ruelle - Bowen measures to describe nonequilibrium steady states, one can in principle compute the coefficients of expansions around equilibrium. We discuss how this can be done in practice, and how the results correspond to the zero noise limit when there is a stochastic perturbation. The approach used is formal rather than rigorous.

One-parameter families of periodic solutions in nonlinear autonomous Hamiltonian systems with two degrees of freedom are considered. Criteria for the existence and stability (checked by the Hessian of the Hamiltonian) as well as bilateral bounds for the periods are obtained. In a system with an even Hamiltonian, the stability is determined by the behaviour of the energy as the parameter increases.

Let f be a sufficiently expanding circle map. We prove that a certain Markov approximation scheme based on a partition of into equal intervals produces a probability measure whose total variation norm distance from the exact absolutely continuous invariant measure is bounded by ; C is a constant depending only on the map f.

We examine the effect of the breaking of vorticity conservation by viscous dissipation on transport in the underlying fluid flow. The transport of interest is between regimes of different characteristic motion and is afforded by the splitting of separatrices. A base flow that is vorticity conserving is therefore assumed to have a separatrix that is either a homoclinic or heteroclinic orbit. The corresponding vorticity dissipating flow, with small time-dependent forcing and viscous parameter , maintains an closeness to the inviscid flow in a weak sense. An appropriate Melnikov theory that allows for such weak perturbations is then developed. A surprisingly simple expression for the leading-order distance between perturbed invariant (stable and unstable) manifolds is derived which depends only on the inviscid flow. Finally, the implications for transport in barotropic jets are discussed.

For a class of two-dimensional hyperbolic maps (which includes certain billiard systems) we construct finite generating partitions. Thus, trajectories of the map can be labelled uniquely by doubly infinite symbol sequences, where the symbols correspond to the atoms of the partition. It is shown that the corresponding conditions are fulfilled in the case of the cardioid billiard, the stadium billiard (and other Bunimovich billiards), planar dispersing and semidispersing billiards.

A structurally stable heteroclinic cycle connecting circles of fixed points of opposite parity is identified in a steady-state mode interaction with symmetry. These fixed points correspond to equilibria in the form of zigzag (odd parity) and varicose (even parity) states. Because of the reflection symmetry this 1:1 mode interaction bears certain similarities to earlier work on the 1:2 mode interaction with symmetry. However, in the present case the cycle connects eight equilibria, four of which are selected from the circle of zigzag equilibria and four from the circle of varicose equilibria. Conditions for the existence and asymptotic stability of the cycle are determined and compared with numerical integration of the mode interaction equations.

A real hyperbolic system is considered that applies near the onset of the oscillatory instability in large spatial domains. The validity of that system requires that some intermediate scales (large compared with the basic wavelength of the unstable modes but small compared with the size of the system) remain inhibited; that condition is analysed in some detail. The dynamics associated with the hyperbolic system is fully analysed to conclude that it is very simple if the coefficient of the cross-nonlinearity is such that , while the system exhibits increasing complexity (including period-doubling sequences, quasiperiodic transitions, crises) as the bifurcation parameter grows if ; if then the system behaves subcritically. Our results are seen to compare well, both qualitatively and quantitatively, with the experimentally obtained ones for the oscillatory instability of straight rolls in pure Rayleigh - Bénard convection.

Recently, Kook and Meiss studied an example of a four-dimensional Lagrangian system and noticed that a minimizer of the Lagrangian action is often hyperbolic. In this paper, we prove that for every homotopy class of , there exists an open set U in the -topology of strictly convex and superlinear Lagrangians L of and such that for each L in U, the -periodic orbits minimizing the Lagrangian action of L in the class have their hyperbolic dimension at most 2.

A general formula for the linearized Poincaré map of a billiard with a potential is derived. Arc length and parallel component of the velocity are shown to be canonical coordinates for the map from bounce to bounce. The stability of periodic orbits is given by the trace of a product of matrices describing the piecewise free motion between reflections and the contributions from the reflections alone. Four billiards with potentials for which the free motion is integrable are treated as examples, the linear gravitational potential, the constant magnetic field, the harmonic potential, and a billiard in a rotating frame of reference, imitating the restricted three-body problem. The linear stability of periodic orbits with periods one and two is analysed with the help of stability diagrams, showing the essential parameter dependence of the residue of the periodic orbits for these examples.

We consider motion in a periodic potential in a classical, quantum, and semiclassical context. Various results on the distribution of asymptotic velocities are proven.