@unpublished{DemircanScheelSeehafer1999,
  author    = {Demircan, Ayhan and Scheel, Stefan and Seehafer, Norbert},
  title     = {Heteroclinic behavior in rotating Rayleigh-B{\´e}nard convection},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14914},
  year      = {1999},
  abstract  = {We investigate numerically the appearance of heteroclinic behavior in a three-dimensional, buoyancy-driven fluid layer with stress-free top and bottom boundaries, a square horizontal periodicity with a small aspect ratio, and rotation at low to moderate rates about a vertical axis. The Prandtl number is 6.8. If the rotation is not too slow, the skewed-varicose instability leads from stationary rolls to a stationary mixed-mode solution, which in turn loses stability to a heteroclinic cycle formed by unstable roll states and connections between them. The unstable eigenvectors of these roll states are also of the skewed-varicose or mixed-mode type and in some parameter regions skewed-varicose like shearing oscillations as well as square patterns are involved in the cycle. Always present weak noise leads to irregular horizontal translations of the convection pattern and makes the dynamics chaotic, which is verified by calculating Lyapunov exponents. In the nonrotating case, the primary rolls lose, depending on the aspect ratio, stability to traveling waves or a stationary square pattern. We also study the symmetries of the solutions at the intermittent fixed points in the heteroclinic cycle.},
  language  = {en}
}
@unpublished{SchmidtmannFeudelSeehafer1997,
  author    = {Schmidtmann, Olaf and Feudel, Fred and Seehafer, Norbert},
  title     = {Nonlinear Galerkin methods for the 3D magnetohydrodynamic equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14431},
  year      = {1997},
  abstract  = {The usage of nonlinear Galerkin methods for the numerical solution of partial differential equations is demonstrated by treating an example. We desribe the implementation of a nonlinear Galerkin method based on an approximate inertial manifold for the 3D magnetohydrodynamic equations and compare its efficiency with the linear Galerkin approximation. Special bifurcation points, time-averaged values of energy and enstrophy as well as Kaplan-Yorke dimensions are calculated for both schemes in order to estimate the number of modes necessary to correctly describe the behavior of the exact solutions.},
  language  = {en}
}
@unpublished{DemircanSeehafer2000,
  author    = {Demircan, Ayhan and Seehafer, Norbert},
  title     = {Nonlinear square patterns in Rayleigh-B{\´e}nard convection},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14986},
  year      = {2000},
  abstract  = {We numerically investigate nonlinear asymmetric square patterns in a horizontal convection layer with up-down reflection symmetry. As a novel feature we find the patterns to appear via the skewed varicose instability of rolls. The time-independent nonlinear state is generated by two unstable checkerboard (symmetric square) patterns and their nonlinear interaction. As the bouyancy forces increase, the interacting modes give rise to bifurcations leading to a periodic alternation between a nonequilateral hexagonal pattern and the square pattern or to different kinds of standing oscillations.},
  language  = {en}
}
@unpublished{FeudelSeehafer1994,
  author    = {Feudel, Fred and Seehafer, Norbert},
  title     = {On the bifurcation phenomena in truncations of the 2D Navier-Stokes equations},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-13390},
  year      = {1994},
  abstract  = {We have studied bifurcation phenomena for the incompressable Navier-Stokes equations in two space dimensions with periodic boundary conditions. Fourier representations of velocity and pressure have been used to transform the original partial differential equations into systems of ordinary differential equations (ODE), to which then numerical methods for the qualitative analysis of systems of ODE have been applied, supplemented by the simulative calculation of solutions for selected initial conditions. Invariant sets, notably steady states, have been traced for varying Reynolds number or strength of the imposed forcing, respectively. A complete bifurcation sequence leading to chaos is described in detail, including the calculation of the Lyapunov exponents that characterize the resulting chaotic branch in the bifurcation diagram.},
  language  = {en}
}
@unpublished{SeehaferSchumacher1998,
  author    = {Seehafer, Norbert and Schumacher, J{\"o}rg},
  title     = {Resistivity profile and instability of the plane sheet pinch},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14686},
  year      = {1998},
  abstract  = {The stability of the quiescent ground state of an incompressible, viscous and electrically conducting fluid sheet, bounded by stress-free parallel planes and driven by an external electric field tangential to the boundaries, is studied numerically. The electrical conductivity varies as cosh-2(x1/a), where x1 is the cross-sheet coordinate and a is the half width of a current layer centered about the midplane of the sheet. For a <~ 0.4L, where L is the distance between the boundary planes, the ground state is unstable to disturbances whose wavelengths parallel to the sheet lie between lower and upper bounds depending on the value of a and on the Hartmann number. Asymmetry of the configuration with respect to the midplane of the sheet, modelled by the addition of an externally imposed constant magnetic field to a symmetric equilibrium field, acts as a stabilizing factor.},
  language  = {en}
}
@unpublished{SeehaferSchumacher1997,
  author    = {Seehafer, Norbert and Schumacher, J{\"o}rg},
  title     = {Squire's theorem for the magnetohydrodynamic sheet pinch},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14628},
  year      = {1997},
  abstract  = {The stability of the quiescent ground state of an incompressible viscous fluid sheet bounded by two parallel planes, with an electrical conductivity varying across the sheet, and driven by an external electric field tangential to the boundaries is considered. It is demonstrated that irrespective of the conductivity profile, as magnetic and kinetic Reynolds numbers (based on the Alfv{\´e}n velocity) are raised from small values, two-dimensional perturbations become unstable first.},
  language  = {en}
}
@unpublished{FeudelSeehaferGalantietal.1996,
  author    = {Feudel, Fred and Seehafer, Norbert and Galanti, Barak and R{\"u}diger, Sten},
  title     = {Symmetry breaking bifurcations for the magnetohydrodynamic equations with helical forcing},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14317},
  year      = {1996},
  abstract  = {We have studied the bifurcations in a three-dimensional incompressible magnetofluid with periodic boundary conditions and an external forcing of the Arnold-Beltrami-Childress (ABC) type. Bifurcation-analysis techniques have been applied to explore the qualitative behavior of solution branches. Due to the symmetry of the forcing, the equations are equivariant with respect to a group of transformations isomorphic to the octahedral group, and we have paid special attention to symmetry-breaking effects. As the Reynolds number is increased, the primary nonmagnetic steady state, the ABC flow, loses its stability to a periodic magnetic state, showing the appearance of a generic dynamo effect; the critical value of the Reynolds number for the instability of the ABC flow is decreased compared to the purely hydrodynamic case. The bifurcating magnetic branch in turn is subject to secondary, symmetry-breaking bifurcations. We have traced periodic and quasi- periodic branches until they end up in chaotic states. In particular detail we have analyzed the subgroup symmetries of the bifurcating periodic branches, which are closely related to the spatial structure of the magnetic field.},
  language  = {en}
}