UK Nonlinear News, November 2003
Conservative dynamics refers to dynamical systems that are measure-preserving. A special class of such systems are Hamiltonian flows and their discrete analog, symplectic maps. Reversibility as well as integrability are symmetry-like properties that, although not necessarily limited to the conservative case, attain special relevance in this context.
Polynomial automorphisms can be counted among the simplest instances of nonlinear maps; yet, as dynamical systems, they may exhibit the full range of complexity from integrability to chaos. This trait makes polynomial automorphisms especially suited to approximate the dynamics of more general systems. In particular we discuss approximation by polynomial shears and by products of Lie transforms. We study reversibility in the group of polynomial automorphisms of the plane and show that in this group reversors always have finite order. We use Jung's theorem and the generalized Hénon maps introduced by Friedland and Milnor to provide normal forms for reversible polynomial automorphisms. We show that every such map is conjugate to a map of the form (h1-1 ... hm-1)r1 (hm ... h1)r0 where each hn is a generalized Hénon transformation and r0, r1 are reversors of a particularly simple form. Some of the dynamical consequences of reversibility are also considered.
Following techniques due to Suris we construct a family of rational diffeomorphisms of R3 that are volume and orientation-preserving and possess a polynomial invariant. Although when restricted to an invariant surface these maps behave locally as area-preserving maps, their dynamics are not equivalent to those of a plane diffeomorphism. We also obtain families of volume-preserving and orientation-reversing diffeomorphisms with an invariant; unlike the orientation-preserving case some of these diffeomorphisms turn out to be polynomial.
Source: James D. Meiss
Repeated reflection of internal waves in enclosed basins may lead to so-called wave attractors. They are studied from a mathematical viewpoint, but also using laboratory experiments. The study is completed with field observations on internal waves in a sea strait.
Internal waves can exist in fluids that are in some way stably stratified: for density-stratified fluids they are called internal gravity waves, for rotating fluids they are called inertial waves. These waves have their maximum amplitude in the interior of the fluid. Their direction of propagation is restricted to a double cone with the direction of gravity or the rotation axis as center line. The opening angle depends only on the wave frequency, and the strength of the stratification or the rotation frequency. This constraint makes the behaviour different from the classical billiard. For a vertical cross-section of a channel with sloping bottom, isolated periodic wave rays may exist to which all other wave rays converge upon repeated reflection. These periodic wave rays are called wave attractors. Contrary to standing waves, they exist for parameter (frequency) intervals. In two dimensions, the spatial pattern of the wave field can be constructed using characteristics (ray-tracing, wave equation in two spatial dimensions). In three dimensions, the pattern is described by the Poincar\'e equation, for which such construction is not yet possible.
First a smooth, convex geometry was studied, which, depending on a parameter, varies between a circle and a triangle. Reflection was studied in terms of iteration of an orientation-preserving map. Windows with attractors were bounded by parameters for which either symmetry or critical points played a special role. No jumps in the Lyapunov exponent were found, contrary to what was found for geometries with corners.
To verify the existence of attractors, laboratory experiments were carried out in a rectangular basin with a sloping boundary on a rotating platform. Two sets of experiments were carried out: one for a large basin and six different forcing frequencies, and one in a smaller basin for three different frequencies. In the smaller tank the three-dimensional aspects of the wave attractors could be studied more easily. Three-dimensional ray-tracing was used to study the possibility of convergence, although no solution of the wave field pattern can be constructed this way.
Finally, internal waves of tidal frequency were studied in the Mozambique Channel, a sea strait between Mozambique and Madagascar, which has steep sloping walls on the scale of internal waves. Field observations were compared with a numerical model. No attractors could be observed. Nevertheless the topography seems to play an important role in the distribution of energy in the channel, despite wave scattering near the surface were the stratification changes rapidly.
A.M.M. Manders, J.J. Duistermaat and L.R.M. Maas. Wave attractors in a smooth convex enclosed geometry; Physica D, in press
A.M.M. Manders and L.R.M. Maas. Observations of inertial waves in a rectangular basin with one sloping boundary. J. Fluid Mech, 493, 59-88, 2003.
Source: Ferdinand Verhulst (F.Verhulst@math.uu.nl).