@unpublished{FeudelSeehafer1995,
  author    = {Feudel, Fred and Seehafer, Norbert},
  title     = {Bifurcations and pattern formation in a 2D Navier-Stokes fluid},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-13907},
  year      = {1995},
  abstract  = {We report on bifurcation studies for the incompressible Navier-Stokes equations in two space dimensions with periodic boundary conditions and an external forcing of the Kolmogorov type. Fourier representations of velocity and pressure have been used to approximate the original partial differential equations by a finite-dimensional system of ordinary differential equations, which then has been studied by means of bifurcation-analysis techniques. A special route into chaos observed for increasing Reynolds number or strength of the imposed forcing is described. It includes several steady states, traveling waves, modulated traveling waves, periodic and torus solutions, as well as a period-doubling cascade for a torus solution. Lyapunov exponents and Kaplan-Yorke dimensions have been calculated to characterize the chaotic branch. While studying the dynamics of the system in Fourier space, we also have transformed solutions to real space and examined the relation between the different bifurcations in Fourier space and toplogical changes of the streamline portrait. In particular, the time-dependent solutions, such as, e.g., traveling waves, torus, and chaotic solutions, have been characterized by the associated fluid-particle motion (Lagrangian dynamics).},
  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@unpublished{FeudelSeehafer1994,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{EngbertScheffczykKrampeetal.1997,
  author    = {Engbert, Ralf and Scheffczyk, Christian and Krampe, Ralf-Thomas and Rosenblum, Mikhael and Kurths, J{\"u}rgen and Kliegl, Reinhold},
  title     = {Tempo-induced transitions in polyrhythmic hand movements},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14380},
  year      = {1997},
  abstract  = {We investigate the cognitive control in polyrhythmic hand movements as a model paradigm for bimanual coordination. Using a symbolic coding of the recorded time series, we demonstrate the existence of qualitative transitions induced by experimental manipulation of the tempo. A nonlinear model with delayed feedback control is proposed, which accounts for these dynamical transitions in terms of bifurcations resulting from variation of the external control parameter. Furthermore, it is shown that transitions can also be observed due to fluctuations in the timing control level. We conclude that the complexity of coordinated bimanual movements results from interactions between nonlinear control mechanisms with delayed feedback and stochastic timing components.},
  language  = {en}
}
@unpublished{DickenMaass1995,
  author    = {Dicken, Volker and Maaß, Peter},
  title     = {Wavelet-Galerkin methods for ill-posed problems},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-13890},
  year      = {1995},
  abstract  = {Projection methods based on wavelet functions combine optimal convergence rates with algorithmic efficiency. The proofs in this paper utilize the approximation properties of wavelets and results from the general theory of regularization methods. Moreover, adaptive strategies can be incorporated still leading to optimal convergence rates for the resulting algorithms. The so-called wavelet-vaguelette decompositions enable the realization of especially fast algorithms for certain operators.},
  language  = {en}
}
@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{BraunFeudelSeehafer1997,
  author    = {Braun, Robert and Feudel, Fred and Seehafer, Norbert},
  title     = {Bifurcations and chaos in an array of forced vortices},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14564},
  year      = {1997},
  abstract  = {We have studied the bifurcation structure of the incompressible two-dimensional Navier-Stokes equations with a special external forcing driving an array of 8×8 counterrotating vortices. The study has been motivated by recent experiments with thin layers of electrolytes showing, among other things, the formation of large-scale spatial patterns. As the strength of the forcing or the Reynolds number is raised the original stationary vortex array becomes unstable and a complex sequence of bifurcations is observed. The bifurcations lead to several periodic branches, torus and chaotic solutions, and other stationary solutions. Most remarkable is the appearance of solutions characterized by structures on spatial scales large compared to the scale of the forcing. We also characterize the different dynamic regimes by means of tracers injected into the fluid. Stretching rates and Hausdorff dimensions of convected line elements are calculated to quantify the mixing process. It turns out that for time-periodic velocity fields the mixing can be very effective.},
  language  = {en}
}
@unpublished{BraunFeudelGuzdar1998,
  author    = {Braun, Robert and Feudel, Fred and Guzdar, Parvez},
  title     = {The route to chaos for a two-dimensional externally driven flow},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14717},
  year      = {1998},
  abstract  = {We have numerically studied the bifurcations and transition to chaos in a two-dimensional fluid for varying values of the Reynolds number. These investigations have been motivated by experiments in fluids, where an array of vortices was driven by an electromotive force. In these experiments, successive changes leading to a complex motion of the vortices, due to increased forcing, have been explored [Tabeling, Perrin, and Fauve, J. Fluid Mech. 213, 511 (1990)]. We model this experiment by means of two-dimensional Navier-Stokes equations with a special external forcing, driving a linear chain of eight counter-rotating vortices, imposing stress-free boundary conditions in the vertical direction and periodic boundary conditions in the horizontal direction. As the strength of the forcing or the Reynolds number is raised, the original stationary vortex array becomes unstable and a complex sequence of bifurcations is observed. Several steady states and periodic branches and a period doubling cascade appear on the route to chaos. For increasing values of the Reynolds number, shear flow develops, for which the spatial scale is large compared to the scale of the forcing. Furthermore, we have investigated the influence of the aspect ratio of the container as well as the effect of no-slip boundary conditions at the top and bottom, on the bifurcation scenario.},
  language  = {en}
}
@unpublished{BraunFeudel1996,
  author    = {Braun, Robert and Feudel, Fred},
  title     = {Supertransient chaos in the two-dimensional complex Ginzburg-Landau equation},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-14099},
  year      = {1996},
  abstract  = {We have shown that the two-dimensional complex Ginzburg-Landau equation exhibits supertransient chaos in a certain parameter range. Using numerical methods this behavior is found near the transition line separating frozen spiral solutions from turbulence. Supertransient chaos seems to be a common phenomenon in extended spatiotemporal systems. These supertransients are characterized by an average transient lifetime which depends exponentially on the size of the system and are due to an underlying nonattracting chaotic set.},
  language  = {en}
}