Table of contents

In the case of the complex Ginzburg-Landau equation in one space dimension it is proven that solutions are completely determined by their values at two sufficiently close points. As a consequence, an upper bound for the winding number of stationary solutions is established in terms of the bifurcation parameters. It is also proven that the fractal dimension of the set of stationary solutions is less than or equal to 4.

As a preliminary step towards understanding the dynamics of the ocean and the impact of the ocean on the global climate system and weather prediction, the authors study the mathematical formulations and attractors of three systems of equations of the ocean, i.e. the primitive equations (the PEs), the primitive equations with vertical viscosity (the PEV²s), and the Boussinesq equations (the BEs), of the ocean. These equations are fundamental equations of the ocean. The BEs are obtained from the general equations of a compressible fluid under the Boussinesq approximation, i.e. the density differences are neglected in the system except in the buoyancy term and in the equation of state. The PEs are derived from the BEs under the hydrostatic approximation for the vertical momentum equation. The PEV²s are the PEs with the viscosity for the vertical velocity retained. This retention is partially based on the important role played by the viscosity in studying the long time behaviour of the ocean, and the Earth's climate.

The authors derive a semiclassical secular equation which applies to quantized (compact) billiards of any shape. Their approach is based on the fact that the billiard boundary defines two dual problems: the 'inside problem' of the bounded dynamics, and the 'outside problem' which can be looked upon as a scattering from the boundary as an obstacle. This duality exists both on the classical and quantum mechanical levels, and is therefore very useful in deriving a semiclassical quantization rule. They obtain a semiclassical secular equation which is based on classical input from a finite number of classical periodic orbits. They compare their result to secular equations which were derived by other means, and provide some numerical data which illustrate their method when applied to the quantization of the Sinai billiard.

Circle packings can be generated by combining hyperbolic and spherical (or Euclidean) tessellation groups. These packings are nonosculatory, and yet complete: an area of full measure is covered by a set of nonoverlapping, nontangent discs. The authors provide the general group-theoretical framework associated to an efficient, geometrically inspired, construction of these packings. They classify the different structures that can be obtained in this way, and they investigate their fractal properties.

The authors give a technique for renormalization of any homeomorphism of the n-torus topologically conjugate to a rotation, and describe a related coding method for orbits of such maps.

The authors examine the dependence of the energy levels of a classically chaotic system on a parameter. They present numerical results which justify the use of a random matrix model for the statistical properties of this dependence. They illustrate the application of their model by calculating both the number of avoided crossings as a function of gap size and the distribution of curvatures of energy levels for a chaotic billiard: the distribution of large curvatures is determined by the density of avoided crossings. Their results confirm that the matrix elements are Gaussian distributed in the semiclassical limit, but they characterize significant deviations from the Gaussian distribution at finite energies.

The author discusses a two-parameter family of truncated sawteeth serving as the asymptotics of a simple scattering model which shows a transition from a regular to a chaotic dynamics. The question as to how the set of trapped trajectories ('the invariant set') evolves with varying family parameters ('the transition problem') proves to be the key to the understanding of this transition. This invariant set follows a hyperbolic cascade of bifurcations of boundary type. At the critical point, where chaos sets in, the invariant set can be represented as the limit set of a generalized cellular automaton (GCA). Alternatively it can be generated from a single seed by (and identified with) the closure of a transformation semi-group of GCAs. This representation can be used to calculate the rotation numbers of cycles belonging to the invariant set. Statistical measures like the topological entropy indicate the occurrence of a phase transition.

The authors consider a class of maps having the origin as a parabolic fixed point with a nondiagonalizable linear part, degenerate in the sense that it has a line of fixed points through it, and they give conditions for the existence and regularity of invariant manifolds. This class is motivated from Poincare maps of flows appearing in celestial mechanics.