Table of contents

The dynamics of parametrically excited surface waves in square containers reveal the effects of symmetry at several levels. In addition to the expected square symmetry D₄ admitted by the fluid equations and the boundary conditions, there are hidden translational and rotational symmetries that further constrain the linear and nonlinear behaviour of the fluid. As a result one finds unexpected degeneracies among the linear wave frequencies and unexpected branches of nonlinear solutions in the bifurcation equations for the surface waves. These additional symmetries are not obvious since they are not symmetries of the square container and consequently do not preserve the boundary conditions of the problem. The author can include them in a theoretical analysis by extending the fluid equations of the original problem to larger domains with greater symmetry; in this enlarged problem the previously hidden symmetries now enter in the usual way. Among other prerequisites, this extension depends on the square container having straight sidewalls.

The authors study the geometrical and statistical structure of a class of coupled map lattices with natural couplings. These are infinite-dimensional analogues of Axiom A systems. Their main result is the existence of a natural spatio-temporal measure which is the spatio-temporal analogue of the SRB measure. They develop a stable manifold theory for such systems as well as spatio-temporal shadowing, Markov partitions and symbolic dynamics. They treat in general terms the question of the existence and uniqueness of Gibbs states for the associated higher-dimensional symbolic systems.

For pt.I see ibid., vol.6, p.165 (1993). The authors study the geometrical and statistical structure of a class of coupled map lattices with natural couplings. These are infinite-dimensional analogues of Axiom A systems. Their main result is the existence of a natural spatio-temporal measure which is the spatio-temporal analogue of the SRB measure. They developed a stable manifold theory for such systems as well as spatio-temporal shadowing, Markov partitions and symbolic dynamics. They treat in general terms the question of the existence and uniqueness of Gibbs states for the associated higher-dimensional symbolic systems.

For ptII see ibid., vol.6, p.201 (1993). The authors study the geometrical and statistical structure of a class of coupled map lattices with natural couplings. These are infinite-dimensional analogues of Axiom A systems. Their main result is the existence of a natural spatio-temporal measure which is the spatio-temporal analogue of the SRB measure. They developed a stable manifold theory for such systems as well as spatio-temporal shadowing, Markov partitions and symbolic dynamics. They treated in general terms the question of the existence and uniqueness of Gibbs states for the associated higher-dimensional symbolic systems. They study the proof of the main theorem which asserts the existence and uniqueness of a natural spatio-temporal measure for certain weakly coupled circle map lattices with a natural coupling.

An inverse scattering 'recipe' is presented for obtaining the solution of a Lagrangian formulation of the Euler equations governing the motion of an unbounded two-dimensional ideal fluid. This formulation is given in terms of a so-called Lax pair of operators. The scattering data are viewed as a delta -data in order to apply the approach for multidimensional inverse scattering. The operator of the Lax pair associated with the spectral problem is treated as a perturbation of the analogous problem for the Laplacian. It is shown that solutions to the direct and inverse problems exist when, in an appropriate sense, this perturbation is small. A particular example is discussed in which the initial velocity has the asymptotic structure of a point vortex. In this case the explicit integral equation is exhibited for the eigenfunctions from which the flow description can be reconstructed.

The authors characterize the chaotic attractors of the Lorenz system associated with R=28 and R=60 in terms of the unstable periodic orbits and their eigenvalues. While the Hausdorff dimension is approximated with very good accuracy in both cases, the topological entropy is computed, in an exact sense only for R=28.

The author proves the existence of nonhyperbolic invariant tori and quasi-periodic saddle-node bifurcations in an unfolding of a double Hopf bifurcation in the presence of a reflection symmetry.

Discrete symmetries of dynamical flows give rise to relations between periodic orbits, reduce the dynamics to a fundamental domain, and lead to factorizations of zeta functions. These factorizations in turn reduce the labour and improve the convergence of cycle expansions for classical and quantum spectra associated with the flow. The general formalism is developed, with the N-disk pinball model used as a concrete example and a series of physically interesting cases worked out in detail.

Geometric monodromy is an obstruction for the global existence of action variables. The authors study two examples which have nontrivial monodromy and exhibit degeneration phenomena. The first is the classical Kirchhoff case of motion of a rigid body in an infinite ideal fluid. The second is a spherical pendulum subject to an axially symmetric quadratic potential.

The authors consider invariant measures for recurrent iterated function systems of conformal maps (which need not be affine). A formula for the singularity spectrum f( alpha ) of the measure is shown to hold in this case. An explicit formula for the Legendre transform of f is also obtained.