Spiral and Antispiral Waves in Reaction-Diffusion Systems

Spiral waves are ubiquitous phenomena in nonlinear chemical, physical, and biological systems. But antispiral waves are infrequent to date. The transition between spiral and antispiral waves has been rarely explored. We have analyzed the extended Brusselator model and the extended Oregonator model by linear stability analysis. We have demonstrated that it is possible and plausible to realize the transition between them by control of diffusion coefficient of inactivator from theoretical analysis and numerical simulations.     

Controlling the transition between Turing and antispiral patterns by using time-delayed-feedback

The controllable transition between Turing and antispiral patterns is studied by using a time-delayed-feedback strategy in a FitzHugh-Nagumo model.We treat the time delay as a perturbation and analyse the effect of the time delay on the Turing and Hopf instabilities near the Turing-Hopf codimension-two phase space.Numerical simulations show that the transition between the Turing patterns(hexagon,stripe,and honeycomb),the dual-mode antispiral,and the antispiral by applying appropriate feedback parameters.The dual-mode antispiral pattern originates from the competition between the Turing and Hopf instabilities.Our results have shown the flexibility of the time delay on controlling the pattern formations near the Turing-Hopf codimension-two phase space.     

Doppler instability of antispiral waves in discrete oscillatory reaction-diffusion media

This paper investigates antispiral wave breakup phenomena in coupled two-dimensional FitzHugh-Nagumo cells with self-sustained oscillation via Hopf bifurcation.When the coupling strength of the active variable decreases to a critical value,wave breakup phenomenon first occurs in the antispiral core region where waves collide with each other and spontaneously break into spatiotemporal turbulence.Measurements reveal for the first time that this breakup phenomenon is due to the mechanism of antispiral Doppler instability.     

反应扩散系统中反螺旋波与反靶波的数值研究

采用三变量Brusselator扩展模型在二维空间对反应扩散系统中反螺旋波和反靶波进行了数值模拟,利用色散关系和参量的时空变化研究了反螺旋波与反靶波的形成机制和时空特性,分析了方程参数对反螺旋波与反靶波的影响,获得了多种不同臂数的反螺旋波.模拟结果表明：反螺旋波源于波失稳、霍普失稳,或两种失稳的共同作用,而在反靶波中除上述两种失稳外还同时存在图灵失稳,波的传播方向均由外向内;反螺旋波波头的相位运动方向与波的走向相同,且旋转周期随臂数的增加逐渐增大;多臂数的反螺旋波由于受微扰及边界条件的影响,在波头的持续旋转运动中可以向臂数少的反螺旋波发生转变,并且在一定条件下单臂反螺旋波可实现到反靶波的转变;当不活跃中间物质的浓度的扩散系数超过临界值时,波的传播方向发生改变,系统可以实现反螺旋波到螺旋波以及反靶波到靶波的转变.     

A kind of extended Korteweg——de Vries equation and solitary wave solutions for interfacial waves in a two-fluid system

This paper considers interfacial waves propagating along the interface between a two-dimensional two-fluid with a flat bottom and a rigid upper boundary. There is a light fluid layer overlying a heavier one in the system, and a small density difference exists between the two layers. It just focuses on the weakly non-linear small amplitude waves by introducing two small independent parameters: the nonlinearity ratio $\varepsilon $, represented by the ratio of amplitude to depth, and the dispersion ratio $\mu $, represented by the square of the ratio of depth to wave length, which quantify the relative importance of nonlinearity and dispersion. It derives an extended KdV equation of the interfacial waves using the method adopted by Dullin {\it et al} in the study of the surface waves when considering the order up to $O(\mu ^2)$. As expected, the equation derived from the present work includes, as special cases, those obtained by Dullin {\it et al} for surface waves when the surface tension is neglected. The equation derived using an alternative method here is the same as the equation presented by Choi and Camassa. Also it solves the equation by borrowing the method presented by Marchant used for surface waves, and obtains its asymptotic solitary wave solutions when the weakly nonlinear and weakly dispersive terms are balanced in the extended KdV equation.     

Wave breaking in Boussinesq models for undular bores

A nonlinear dispersive model equation is used to study the onset of breaking in long waves behind the front of an undular bore. According to experiments conducted by Favre (1935) [1], weak bores have a smooth, but oscillatory structure, with undulations appearing behind the bore front. With increasing bore strength, the amplitude of these oscillations grows until one or several of them start breaking. The change in type from the purely undular bore occurs at a sharply defined depth ratio which is under review in this article. A convective breaking criterion is put forward, and numerical computations are used to compare the predictions of this model to Favre?s wavetank experiments. It appears that the numerical results underpredict the appearance of breaking waves, but are in good qualitative agreement with the experiments. The results are interpreted with the aid of exact solitary-wave solutions, and it is found that the transition from purely undular to breaking bore may be recast with the help of a breaking criterion for solitary waves.     

两种扩展Harper模型的波包动力学

本文通过二次矩M₂(t)和概率分布W_n(t)数值地研究了两种扩展Harper模型的波包动力学,得到了这两种模型中各个相、各条临界线以及三相点的波包扩散情况.对于第一种扩展Harper模型,发现两个金属相中波包是弹道扩散的,在绝缘体相中波包不扩散,而在三相点以及各条临界线上波包是反常扩散的.同时,发现金属相—金属相转变的临界线上的波包动力学行为与金属相—绝缘体相转变的临界线上的相同,但三相点的动力学行为与各临 关键词： 金属绝缘体转变 扩展Harper模型 波包动力学     

Pattern formation in superdiffusion Oregonator model

Pattern formations in an Oregonator model with superdiffusion are studied in two-dimensional(2D) numerical simulations. Stability analyses are performed by applying Fourier and Laplace transforms to the space fractional reaction–diffusion systems. Antispiral, stable turing patterns, and travelling patterns are observed by changing the diffusion index of the activator. Analyses of Floquet multipliers show that the limit cycle solution loses stability at the wave number of the primitive vector of the travelling hexagonal pattern. We also observed a transition between antispiral and spiral by changing the diffusion index of the inhibitor.     

Complex oscillations and waves of calcium in pancreatic acinar cells

We perform a bifurcation analysis of a model of Ca²⁺ wave propagation in the basal region of pancreatic acinar cells. The model we consider was first presented in Sneyd et al. [J. Sneyd, K. Tsaneva-Atanasova, J.I.E. Bruce, S.V. Straub, D.R. Giovannucci, D.I. Yule, A model of calcium waves in pancreatic and parotid acinar cells, Biophys. J. 85 (2003) 1392–1405], where a partial bifurcation analysis was given of the model in the absence of diffusion. We obtain more complete information about bifurcations of the diffusionless model via numerical studies, then analyse the spatially extended model by numerical investigation of the travelling wave equations and direct numerical solution of the model equations. We find solitary waves in the model equations arising from homoclinic bifurcations in the travelling wave equations. The solitary waves exist and appear to be stable for a significant interval of the primary bifurcation parameter (i.e., the concentration of inositol trisphosphate) but are eventually replaced by irregular spatio-temporal behaviour. The homoclinic bifurcations are related to a number of complicated mathematical structures in the travelling wave equations, including an anomalous homoclinic-Hopf bifurcation, heteroclinic bifurcations between an equilibrium and a periodic orbit, and homoclinic bifurcations of periodic orbits.