@article{BordyugovFischerEngeletal.2010,
  author    = {Bordyugov, Grigory and Fischer, Nils and Engel, Harald and Manz, Niklas and Steinbock, Oliver},
  title     = {Anomalous dispersion in the Belousov-Zhabotinsky reaction : experiments and modeling},
  issn      = {0167-2789},
  doi       = {10.1016/j.physd.2009.10.022},
  year      = {2010},
  abstract  = {We report results on dispersion relations and instabilities of traveling waves in excitable systems. Experiments employ solutions of the 1,4-cyclohexanedione Belousov-Zhabotinsky reaction confined to thin capillary tubes which create a pseudo-one-dimensional system. Theoretical analyses focus on a three-variable reaction-diffusion model that is known to reproduce qualitatively many of the experimentally observed dynamics. Using continuation methods, we show that the transition from normal, monotonic to anomalous, single-overshoot dispersion curves is due to an orbit flip bifurcation of the solitary pulse homoclinics. In the case of "wave stacking", this anomaly induces attractive pulse interaction, slow solitary pulses, and faster wave trains. For "wave merging", wave trains break up in the wake of the slow solitary pulse due to an instability of wave trains at small wavelength. A third case, "wave tracking" is characterized by the non-existence of solitary waves but existence of periodic wave trains. The corresponding dispersion curve is a closed curve covering a finite band of wavelengths.},
  language  = {en}
}
@misc{TotzLoeberTotzetal.2018,
  author    = {Totz, Sonja Juliana and L{\"o}ber, Jakob and Totz, Jan Frederik and Engel, Harald},
  title     = {Control of transversal instabilities in reaction-diffusion systems},
  series = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  journal   = {Postprints der Universit{\"a}t Potsdam : Mathematisch-Naturwissenschaftliche Reihe},
  number    = {962},
  issn      = {1866-8372},
  doi       = {10.25932/publishup-46976},
  url       = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-469762},
  pages     = {19},
  year      = {2018},
  abstract  = {In two-dimensional reaction-diffusion systems, local curvature perturbations on traveling waves are typically damped out and vanish. However, if the inhibitor diffuses much faster than the activator, transversal instabilities can arise, leading from flat to folded, spatio-temporally modulated waves and to spreading spiral turbulence. Here, we propose a scheme to induce or inhibit these instabilities via a spatio-temporal feedback loop. In a piecewise-linear version of the FitzHugh-Nagumo model, transversal instabilities and spiral turbulence in the uncontrolled system are shown to be suppressed in the presence of control, thereby stabilizing plane wave propagation. Conversely, in numerical simulations with the modified Oregonator model for the photosensitive Belousov-Zhabotinsky reaction, which does not exhibit transversal instabilities on its own, we demonstrate the feasibility of inducing transversal instabilities and study the emerging wave patterns in a well-controlled manner.},
  language  = {en}
}
@article{TotzLoeberTotzetal.2018,
  author    = {Totz, Sonja Juliana and L{\"o}ber, Jakob and Totz, Jan Frederik and Engel, Harald},
  title     = {Control of transversal instabilities in reaction-diffusion systems},
  series = {New journal of physics : the open-access journal for physics},
  volume    = {20},
  journal   = {New journal of physics : the open-access journal for physics},
  publisher = {IOP Publ. Ltd.},
  address   = {Bristol},
  issn      = {1367-2630},
  doi       = {10.1088/1367-2630/aabce5},
  pages     = {16},
  year      = {2018},
  abstract  = {In two-dimensional reaction-diffusion systems, local curvature perturbations on traveling waves are typically damped out and vanish. However, if the inhibitor diffuses much faster than the activator, transversal instabilities can arise, leading from flat to folded, spatio-temporally modulated waves and to spreading spiral turbulence. Here, we propose a scheme to induce or inhibit these instabilities via a spatio-temporal feedback loop. In a piecewise-linear version of the FitzHugh-Nagumo model, transversal instabilities and spiral turbulence in the uncontrolled system are shown to be suppressed in the presence of control, thereby stabilizing plane wave propagation. Conversely, in numerical simulations with the modified Oregonator model for the photosensitive Belousov-Zhabotinsky reaction, which does not exhibit transversal instabilities on its own, we demonstrate the feasibility of inducing transversal instabilities and study the emerging wave patterns in a well-controlled manner.},
  language  = {en}
}
@article{ZykovBordyugovLentzetal.2010,
  author    = {Zykov, Vladimir and Bordyugov, Grigory and Lentz, Hartmut and Engel, Harald},
  title     = {Hysteresis phenomenon in the dynamics of spiral waves rotating around a hole},
  issn      = {0167-2789},
  doi       = {10.1016/j.physd.2009.07.018},
  year      = {2010},
  abstract  = {Hysteresis in the pinning-depinning transitions of spiral waves rotating around a hole in a circular shaped two- dimensional excitable medium is studied both by use of the continuation software AUTO and by direct numerical integration of the reaction-diffusion equations for the FitzHugh-Nagumo model. In order to clarify the role of different factors in this phenomenon, a kinematical description is applied. It is found that the hysteresis phenomenon computed for the reaction-diffusion model can be reproduced qualitatively only when a nonlinear eikonal equation (i.e. velocity- curvature relationship) is assumed. However, to obtain quantitative agreement, the dispersion relation has to be taken into account.},
  language  = {en}
}