TY - JOUR A1 - Zykov, Vladimir A1 - Bordyugov, Grigory A1 - Lentz, Hartmut A1 - Engel, Harald T1 - Hysteresis phenomenon in the dynamics of spiral waves rotating around a hole N2 - 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. Y1 - 2010 UR - http://www.sciencedirect.com/science/journal/01672789 U6 - https://doi.org/10.1016/j.physd.2009.07.018 SN - 0167-2789 ER - TY - GEN A1 - Totz, Sonja Juliana A1 - Löber, Jakob A1 - Totz, Jan Frederik A1 - Engel, Harald T1 - Control of transversal instabilities in reaction-diffusion systems T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - 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. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 962 KW - traveling waves KW - control KW - transversal instabilities Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-469762 SN - 1866-8372 IS - 962 ER - TY - JOUR A1 - Totz, Sonja Juliana A1 - Löber, Jakob A1 - Totz, Jan Frederik A1 - Engel, Harald T1 - Control of transversal instabilities in reaction-diffusion systems JF - New journal of physics : the open-access journal for physics N2 - 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. KW - traveling waves KW - control KW - transversal instabilities Y1 - 2018 U6 - https://doi.org/10.1088/1367-2630/aabce5 SN - 1367-2630 VL - 20 PB - IOP Publ. Ltd. CY - Bristol ER - TY - JOUR A1 - Bordyugov, Grigory A1 - Fischer, Nils A1 - Engel, Harald A1 - Manz, Niklas A1 - Steinbock, Oliver T1 - Anomalous dispersion in the Belousov-Zhabotinsky reaction : experiments and modeling N2 - 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. Y1 - 2010 UR - http://www.sciencedirect.com/science/journal/01672789 U6 - https://doi.org/10.1016/j.physd.2009.10.022 SN - 0167-2789 ER -