@unpublished{BraunFeudelSeehafer1997,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}
}
@article{BraunFeudelSeehafer1997,
  author    = {Braun, Robert and Feudel, Fred and Seehafer, Norbert},
  title     = {Bifurcations and chaos in an array of forced vortices},
  year      = {1997},
  language  = {en}
}
@book{BraunFeudelSeehafer1997,
  author    = {Braun, Robert and Feudel, Fred and Seehafer, Norbert},
  title     = {Bifurcations and chaos in an array of forced vortices},
  series = {Preprint NLD},
  volume    = {37},
  journal   = {Preprint NLD},
  publisher = {Univ. Potsdam},
  address   = {Potsdam},
  issn      = {1432-2935},
  pages     = {7 S. : graph. Darst.},
  year      = {1997},
  language  = {en}
}
@article{BrownCanfieldFieldetal.1999,
  author    = {Brown, M. R. and Canfield, R. C. and Field, G. and Kulsrud, R. and Pevtsov, A. A. and Rosner, R. and Seehafer, Norbert},
  title     = {Magnetic helicity in space and laboratory plasmas: Editorial summary},
  year      = {1999},
  language  = {en}
}
@article{DemircanScheelSeehafer2000,
  author    = {Demircan, Ayhan and Scheel, S. and Seehafer, Norbert},
  title     = {Heteroclinic behavior in rotating Rayleigh-Benard convection},
  year      = {2000},
  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{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}
}
@article{DemircanSeehafer2001,
  author    = {Demircan, Ayhan and Seehafer, Norbert},
  title     = {Dynamos in rotating and nonrotating convection in the form of asymmetric squares},
  year      = {2001},
  abstract  = {We study the dynamo properties of asymmetric square patterns in Boussinesq Rayleigh-B'enard convection in a plane horizontal layer. Cases without rotation and with weak rotation about a vertical axis are considered. There exist different types of solutions distinguished by their symmetry, among them such with flows possessing a net helicity and being capable of kinematic dynamo action in the presence as well as in the absence of rotation. In the nonrotating case these flows are, however, always only kinematic, not nonlinear dynamos. Nonlinearly the back-reaction of the magnetic field then forces the solution into the basin of attraction of a roll pattern incapable of dynamo action. But with rotation added parameter regions are found where the Coriolis force counteracts the Lorentz force in such a way that the asymmetric squares are also nonlinear dynamos.},
  language  = {en}
}
@article{DemircanSeehafer2001,
  author    = {Demircan, Ayhan and Seehafer, Norbert},
  title     = {Nonlinear square patterns in Rayleigh-Benard convection},
  year      = {2001},
  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}
}
@article{DemircanSeehafer2002,
  author    = {Demircan, Ayhan and Seehafer, Norbert},
  title     = {Dynamo in asymmetric square convection},
  issn      = {0309-1929},
  year      = {2002},
  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}
}
@article{DemircanSeehafer2000,
  author    = {Demircan, Ayhan and Seehafer, Norbert},
  title     = {Heteroclinic behavior in rotating Rayleigh-B{\´e}nard convection},
  year      = {2000},
  abstract  = {We investigate numerically the appearance of heteroclinic behavior in a three-dimensional, buoyancy-driven, rotating fluid layer. Periodic boundary conditions in the horizontal directions and stress-free boundary conditions at the top and bottom are assumed.},
  language  = {en}
}
@article{DemircanSeehafer1999,
  author    = {Demircan, Ayhan and Seehafer, Norbert},
  title     = {Bifurcation to oscillations and chaos in rotating convection},
  year      = {1999},
  language  = {en}
}
@article{DonnerFeudelSeehaferetal.2007,
  author    = {Donner, Reik Volker and Feudel, Fred and Seehafer, Norbert and Sanjuan, Miguel Angel Fernandez},
  title     = {Hierarchical modeling of a forced Roberts Dynamo},
  issn      = {0218-1274},
  doi       = {10.1142/S021812740701941X},
  year      = {2007},
  abstract  = {We investigate the dynamo effect in a flow configuration introduced by G. O. Roberts in 1972. Based on a clear energetic hierarchy of Fourier components on the steady-state dynamo branch, an approximate model of interacting modes is constructed covering all essential features of the complete system but allowing simulations with a minimum amount of computation time. We use this model to study the excitation mechanism of the dynamo, the transition from stationary to time-dependent dynamo solutions and the characteristic properties of the latter ones.},
  language  = {en}
}
@article{DonnerSeehaferSanjuanetal.2006,
  author    = {Donner, Reik Volker and Seehafer, Norbert and Sanjuan, Miguel Angel Fernandez and Feudel, Fred},
  title     = {Low-dimensional dynamo modelling and symmetry-breaking bifurcations},
  series = {Physica. D, Nonlinear phenomena},
  volume    = {223},
  journal   = {Physica. D, Nonlinear phenomena},
  number    = {2},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0167-2789},
  doi       = {10.1016/j.physd.2006.08.022},
  pages     = {151 -- 162},
  year      = {2006},
  abstract  = {Motivated by the successful Karlsruhe dynamo experiment, a relatively low-dimensional dynamo model is proposed. It is based on a strong truncation of the magnetohydrodynamic (MHD) equations with an external forcing of the Roberts type and the requirement that the model system satisfies the symmetries of the full MHD system, so that the first symmetry-breaking bifurcations can be captured. The backbone of the Roberts dynamo is formed by the Roberts flow, a helical mean magnetic field and another part of the magnetic field coupled to these two by triadic mode interactions. A minimum truncation model (MTM) containing only these energetically dominating primary mode triads is fully equivalent to the widely used first-order smoothing approximation. However, it is shown that this approach works only in the limit of small wave numbers of the excited magnetic field or small magnetic Reynolds numbers (\$Rm ll 1\$). To obtain dynamo action under more general conditions, secondary mode},
  language  = {en}
}
@article{FeudelGellertRuedigeretal.2003,
  author    = {Feudel, Fred and Gellert, Marcus and R{\"u}diger, Sten and Witt, Annette and Seehafer, Norbert},
  title     = {Dynamo effect in a driven helical flow},
  year      = {2003},
  language  = {en}
}
@article{FeudelRuedigerSeehafer2001,
  author    = {Feudel, Fred and R{\"u}diger, Sten and Seehafer, Norbert},
  title     = {Bifurcation phenomena and dynamo effect in electrically conducting fluids},
  year      = {2001},
  abstract  = {Electrically conducting fluids in motion can act as self-excited dynamos. The magnetic fields of celestial bodies like the Earth and the Sun are generated by such dynamos. Their theory aims at modeling and understanding both the kinematic and dynamic aspects of the underlying processes. Kinematic dynamo models, in which for a prescribed flow the linear induction equation is solved and growth rates of the magnetic field are calculated, have been studied for many decades. But in order to get consistent models and to take into account the back-reaction of the magnetic field on the fluid motion, the full nonlinear system of the magnetohydrodynamic (MHD) equations has to be studied. It is generally accepted that these equations, i.e. the Navier-Stokes equation (NSE) and the induction equation, provide a theoretical basis for the explanation of the dynamo effect. The general idea is that mechanical energy pumped into the fluid by heating or other mechanisms is transferred to the magnetic field by nonlinear interactions. For two special helical flows which are known to be effective kinematic dynamos and which can be produced by appropriate external mechanical forcing, we review the nonlinear dynamo properties found in the framework of the full MHD equations. Specifically, we deal with the ABC flow (named after Arnold, Beltrami and Childress) and the Roberts flow (after G.~O. Roberts). The appearance of generic dynamo effects is demonstrated. Applying special numerical bifurcation-analysis techniques to high-dimensional approximations in Fourier space and varying the Reynolds number (or the strength of the forcing) as the relevant control parameter, qualitative changes in the dynamics are investigated. We follow the bifurcation sequences until chaotic states are reached. The transitions from the primary flows with vanishing magnetic field to dynamo-active states are described in particular detail. In these processes the stagnation points of the flows and their heteroclinic connections play a promoting role for the magnetic field generation. By the example of the Roberts flow we demonstrate how the break up of the heteroclinic lines after the primary bifurcation leads to a complicated intersection of stable and unstable manifolds forming a chaotic web which is in turn correlated with the spatial appearance of the dynamo.},
  language  = {en}
}
@article{FeudelSeehafer1995,
  author    = {Feudel, Fred and Seehafer, Norbert},
  title     = {On the bifurcation phenomena in truncations of the 2D Navier-Stokes equations},
  year      = {1995},
  language  = {en}
}
@article{FeudelSeehafer1995,
  author    = {Feudel, Fred and Seehafer, Norbert},
  title     = {Bifurcations and pattern formation in two-dimensional Navier-Stokes fluid},
  year      = {1995},
  language  = {en}
}
@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}
}