@article{FeudelWittGellertetal.2005,
  author    = {Feudel, Fred and Witt, Annette and Gellert, Marcus and Kurths, J{\"u}rgen and Grebogi, Celso and Sanjuan, Miguel Angel Fernandez},
  title     = {Intersections of stable and unstable manifolds : the skeleton of Lagrangian chaos},
  year      = {2005},
  abstract  = {We study Hamiltonian chaos generated by the dynamics of passive tracers moving in a two-dimensional fluid flow and describe the complex structure formed in a chaotic layer that separates a vortex region from the shear flow. The stable and unstable manifolds of unstable periodic orbits are computed. It is shown that their intersections in the Poincare map as an invariant set of homoclinic points constitute the backbone of the chaotic layer. Special attention is paid to the finite time properties of the chaotic layer. In particular, finite time Lyapunov exponents are computed and a scaling law of the variance of their distribution is derived. Additionally, the box counting dimension as an effective dimension to characterize the fractal properties of the layer is estimated for different duration times of simulation. Its behavior in the asymptotic time limit is discussed. By computing the Lyapunov exponents and by applying methods of symbolic dynamics, the formation of the layer as a function of the external forcing strength, which in turn represents the perturbation of the originally integrable system, is characterized. In particular, it is shown that the capture of KAM tori by the layer has a remarkable influence on the averaged Lyapunov exponents. (C) 2004 Elsevier Ltd. All rights reserved},
  language  = {en}
}
@article{FeudelBergemannTuckermanetal.2011,
  author    = {Feudel, Fred and Bergemann, Kay and Tuckerman, Laurette S. and Egbers, C. and Futterer, B. and Gellert, Marcus and Hollerbach, Rainer},
  title     = {Convection patterns in a spherical fluid shell},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {83},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {4},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.83.046304},
  pages     = {8},
  year      = {2011},
  abstract  = {Symmetry-breaking bifurcations have been studied for convection in a nonrotating spherical shell whose outer radius is twice the inner radius, under the influence of an externally applied central force field with a radial dependence proportional to 1/r(5). This work is motivated by the GeoFlow experiment, which is performed under microgravity condition at the International Space Station where this particular central force can be generated. In order to predict the observable patterns, simulations together with path-following techniques and stability computations have been applied. Branches of axisymmetric, octahedral, and seven-cell solutions have been traced. The bifurcations producing them have been identified and their stability ranges determined. At higher Rayleigh numbers, time-periodic states with a complex spatiotemporal symmetry are found, which we call breathing patterns.},
  language  = {en}
}
@article{SeehaferGellertKuzanyanetal.2003,
  author    = {Seehafer, Norbert and Gellert, Marcus and Kuzanyan, Kirill M. and Pipin, V. V.},
  title     = {Helicity and the solar dynamo},
  year      = {2003},
  language  = {en}
}
@article{FeudelTuckermanGellertetal.2015,
  author    = {Feudel, Fred and Tuckerman, L. S. and Gellert, Marcus and Seehafer, Norbert},
  title     = {Bifurcations of rotating waves in rotating spherical shell convection},
  series = {Physical Review E},
  volume    = {92},
  journal   = {Physical Review E},
  number    = {5},
  publisher = {American Physical Society},
  address   = {Woodbury},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.92.053015},
  year      = {2015},
  abstract  = {The dynamics and bifurcations of convective waves in rotating and buoyancy-driven spherical Rayleigh-Benard convection are investigated numerically. The solution branches that arise as rotating waves (RWs) are traced by means of path-following methods, by varying the Rayleigh number as a control parameter for different rotation rates. The dependence of the azimuthal drift frequency of the RWs on the Ekman and Rayleigh numbers is determined and discussed. The influence of the rotation rate on the generation and stability of secondary branches is demonstrated. Multistability is typical in the parameter range considered.},
  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}
}
@phdthesis{Gellert2004,
  author    = {Gellert, Marcus},
  title     = {Zum Dynamoeffekt in extern getriebenen Str{\"o}mungen},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-0001705},
  school      = {Universit{\"a}t Potsdam},
  year      = {2004},
  abstract  = {Die Frage nach der Herkunft und der dynamischen Entwicklung langlebiger kosmischer Magnetfelder ist in vielen Details noch unbeantwortet. Es besteht zwar kein Zweifel daran, dass das Magnetfeld der Erde und anderer kosmischer Objekte durch den sogenannten Dynamoeffekt verursacht werden, der genaue Mechanismus als auch die notwendigen Voraussetzungen und Randbedingungen der zugrundeliegenden Str{\"o}mungen sind aber weitgehend unbekannt. Die f{\"u}r einen Dynamo interessanten Str{\"o}mungsmuster, die im Inneren von Himmelsk{\"o}rpern durch Konvektion und differentielle Rotation entstehen, sind Konvektionsrollen parallel zur Rotationsachse. Auf einer Str{\"o}mung mit eben solcher Geometrie, der sogenannten Roberts-Str{\"o}mung, basieren die in der vorliegenden Arbeit untersuchten Dynamomodelle. Mit Methoden der nichtlinearen Dynamik wird versucht, das Systemverhalten bei {\"A}nderung der Systemparamter genauer zu charakterisieren. Die numerischen Untersuchungen beginnen mit einer Analyse der Dynamoaktivit{\"a}t der Roberts-Str{\"o}mung in Abh{\"a}ngigkeit von den zwei freien Parametern in den Modellgleichungen, der magnetischen Prandtl-Zahl und der St{\"a}rke des Energieinputs. Gefunden werden verschiedene L{\"o}sungstypen die von einem station{\"a}ren Magnetfeld {\"u}ber periodische bis zu chaotischen Zust{\"a}nden reichen. Die yugrundeliegenden Symmetrien werden beschrieben und die Bifurkationen, die zum Wechsel der L{\"o}sungstypen f{\"u}hren, charakterisiert. Zus{\"a}tzlich gibt es Bereiche bei sehr kleinen Prandtl-Zahlen, in denen {\"u}berhaupt kein Dynamo existiert. Dieses Verhalten wird in der Literatur auch f{\"u}r viele andere numerisch ausgewertete Modelle beschrieben. Im {\"U}bergangsbereich zwischen dynamoaktivem und dynamoinaktivem Bereich wird das Auftreten einer sogenannten Blowout-Bifurkation gefunden. Desweiteren besch{\"a}ftigt sich die Arbeit mit der Frage, inwiefern Helizit{\"a}t, also eine schraubenf{\"o}rmige Bewegung, der Str{\"o}mung den Dynamoeffekt beeinflusst. Dazu werden {\"a}hnliche Str{\"o}mungstypen verglichen, die sich haupts{\"a}chlich in ihrem Helizit{\"a}tswert unterscheiden. Es wird gefunden, dass ein bestimmter Wert der Helizit{\"a}t nicht unterschritten werden darf, um einen stabilen Roberts-Dynamo zu erhalten.},
  language  = {de}
}
@article{FeudelSeehaferTuckermanetal.2013,
  author    = {Feudel, Fred and Seehafer, Norbert and Tuckerman, Laurette S. and Gellert, Marcus},
  title     = {Multistability in rotating spherical shell convection},
  series = {Physical review : E, Statistical, nonlinear and soft matter physics},
  volume    = {87},
  journal   = {Physical review : E, Statistical, nonlinear and soft matter physics},
  number    = {2},
  publisher = {American Physical Society},
  address   = {College Park},
  issn      = {1539-3755},
  doi       = {10.1103/PhysRevE.87.023021},
  pages     = {8},
  year      = {2013},
  abstract  = {The multiplicity of stable convection patterns in a rotating spherical fluid shell heated from the inner boundary and driven by a central gravity field is presented. These solution branches that arise as rotating waves (RWs) are traced for varying Rayleigh number while their symmetry, stability, and bifurcations are studied. At increased Rayleigh numbers all the RWs undergo transitions to modulated rotating waves (MRWs) which are classified by their spatiotemporal symmetry. The generation of a third frequency for some of the MRWs is accompanied by a further loss of symmetry. Eventually a variety of MRWs, three-frequency solutions, and chaotic saddles and attractors control the dynamics for higher Rayleigh numbers.},
  language  = {en}
}