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Register a new userExpanding scroll rings in a model for the photosensitive Belousov-Zhabotinsky reaction
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While free scroll rings are non-stationary objects that either grow or contract with time, spatial confinement can have a large impact on their evolution reaching from significant lifetime extension [J. F. Totz , H. Engel, and O. Steinbock, New J. Phys. 17, 093043 (2015)] up to formation of stable stationary and breathing pacemakers [A. Azhand, J. F. Totz, and H. Engel, Europhys. Lett. 108, 10004 (2014)]. Here, we explore the parameter range in which the interaction between an axis-symmetric scroll ring and a confining planar no-flux boundary can be studied experimentally in transparent gel layers supporting chemical wave propagation in the photosensitive variant of the Belousov-Zhabotinsky medium. Based on full three-dimensional simulations of the underlying modified complete Oregonator model for experimentally realistic parameters, we determine the conditions for successful initiation of scroll rings in a phase diagram spanned by the layer thickness and the applied light intensity. Furthermore, we discuss whether the illumination-induced excitability gradient due to Lambert-Beers law as well as a possible inclination of the filament plane with respect to the no-flux boundary can destabilize the scroll ring.
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Three-dimensional excitable systems can selforganize vortex patterns that rotate around one-dimensional phase singularities called filaments. In experiments with the Belousov-Zhabotinsky reaction and numerical simulations, we pin these scroll waves to moving heterogeneities and demonstrate the controlled repositioning of their rotation centers. If the pinning site extends only along a portion of the filament, the phase singularity is stretched out along the trajectory of the heterogeneity which effectively writes the singularity into the system. Its trailing end point follows the heterogeneity with a lower velocity. This velocity, its dependence on the placement of the anchor, and the shape of the filament are explained by a curvature flow model.
Three-dimensional excitable systems can create nonlinear scroll waves that rotate around one-dimensional phase singularities. Recent theoretical work predicts that these filaments drift along step-like height variations. Here we test this prediction using experiments with thin layers of the Belousov-Zhabotinsky reaction. We observe that over short distances scroll waves are attracted towards the step and then rapidly commence a steady drift along the step line. The translating filaments always reside in the shallow subsystem and terminate on the step plateau near the edge. Accordingly filaments in the deep subsystem initially collide with and shorten at the step wall. The drift speeds obey the predicted proportional dependence on the logarithm of the height ratio and the direction depends on the vortex chirality. We also observe drift along the perimeter of rectangular plateaus and find that the filaments perform sharp turns at the corners. In addition, we investigate rectangular troughs for which vortices of equal chirality can drift in different directions. The latter two effects are reproduced in numerical simulations with the Barkley model. The simulations show that narrow troughs instigate scroll wave encounters that induce repulsive interaction and symmetry breaking. Similar phenomena could exist in the geometrical complicated ventricles of the human heart where reentrant vortex waves cause tachycardia and fibrillation.
In this paper the travelling wave solutions in the adiabatic model with two-step chain branching reaction mechanism are investigated both numerically and analytically in the limit of equal diffusivity of reactant, radicals and heat. The properties of these solutions and their stability are investigated in detail. The behaviour of combustion waves are demonstrated to have similarities with the properties of nonadiabatic one-step combustion waves in that there is a residual amount of fuel left behind the travelling waves and the solutions can exhibit extinction. The difference between the nonadiabatic one-step and adiabatic two-step models is found in the behaviour of the combustion waves near the extinction condition. It is shown that the flame velocity drops down to zero and a standing combustion wave is formed as the extinction condition is reached. Prospects of further work are also discussed.
The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249--1267]. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Singular perturbation techniques are used to construct steady-state hot-spot patterns in one and two-dimensional spatial domains, and new types of nonlocal eigenvalue problems are derived that determine the stability of these hot-spot patterns to ${mathcal O}(1)$ time-scale instabilities. From an analysis of these nonlocal eigenvalue problems, a critical threshold $K_c$ is determined such that a pattern consisting of $K$ hot-spots is unstable to a competition instability if $K>K_c$. This instability, due to a positive real eigenvalue, triggers the collapse of some of the hot-spots in the pattern. Furthermore, in contrast to the well-known stability results for spike patterns of the Gierer-Meinhardt reaction-diffusion model, it is shown for the crime model that there is only a relatively narrow parameter range where oscillatory instabilities in the hot-spot amplitudes occur. Such an instability, due to a Hopf bifurcation, is studied explicitly for a single hot-spot in the shadow system limit, for which the diffusivity of criminals is asymptotically large. Finally, the parameter regime where localized hot-spots occur is compared with the parameter regime, studied in previous works, where Turing instabilities from a spatially uniform steady-state occur.
In this paper, we study expanding phenomena in the setting of matrix rings. More precisely, we will prove that If $A$ is a set of $M_2(mathbb{F}_q)$ and $|A|gg q^{7/2}$, then we have [|A(A+A)|, ~|A+AA|gg q^4.] If $A$ is a set of $SL_2(mathbb{F}_q)$ and $|A|gg q^{5/2}$, then we have [|A(A+A)|, ~|A+AA|gg q^4.] We also obtain similar results for the cases of $A(B+C)$ and $A+BC$, where $A, B, C$ are sets in $M_2(mathbb{F}_q)$.
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