Spiral wave chimeras in populations of oscillators coupled to a slowly varying diffusive environment

Chimera states are firstly discovered in nonlocally coupled oscillator systems. Such a nonlocal coupling arises typically as oscillators are coupled via an external environment whose characteristic time scale τ is so small (i.e., τ → 0) that it could be eliminated adiabatically. Nevertheless, whether the chimera states still exist in the opposite situation (i.e., τ ≫ 1) is unknown. Here, by coupling large populations of Stuart−Landau oscillators to a diffusive environment, we demonstrate that spiral wave chimeras do exist in this oscillator-environment coupling system even when τ is very large. Various transitions such as from spiral wave chimeras to spiral waves or unstable spiral wave chimeras as functions of the system parameters are explored. A physical picture for explaining the formation of spiral wave chimeras is also provided. The existence of spiral wave chimeras is further confirmed in ensembles of FitzHugh−Nagumo oscillators with the similar oscillator-environment coupling mechanism. Our results provide an affirmative answer to the observation of spiral wave chimeras in populations of oscillators mediated via a slowly changing environment and give important hints to generate chimera patterns in both laboratory and realistic chemical or biological systems.     

反应扩散方程的连续时空有限元方法

本文研究了反应扩散方程的连续时空有限元方法.首先建立了其连续时空有限元格式并证明了有限元解的存在唯一性及稳定性.然后通过引入时空投影算子在没有时空网格限制的条件下给出其近似解在节点处的L~2,H~1最优范数估计以及全局L~2(L~2),L~2(H~1)最优范数估计.最后给出两个数值算例来验证方法的有效性与灵活性并说明结论的正确性.     

Traveling wave solutions in the mathematical model of blood coagulation

Under normal conditions, blood coagulation provides an effective protective mechanism preventing bleeding in case of vessel damage. Details of its functioning are of particular importance since any blood coagulation disorders lead to severe physiological aggravations. Multiple experimental and computational studies demonstrate the thrombin concentration distribution to determine the spatio-temporal dynamics of clot formation. Propagating from the injury site with constant speed, thrombin concentration profile can be modeled with a traveling wave solution of the system of partial differential equations describing main reactions of the coagulation cascade. In the current study, we derive conditions on the existence and stability of such solutions and provide an analytic approach of their wave speed estimation.     

Approximation of systems of singularly perturbed elliptic reaction-diffusion equations with two parameters

In a rectangle, the Dirichlet problem for a system of two singularly perturbed elliptic reaction-diffusion equations is considered. The higher order derivatives of the ith equation are multiplied by the perturbation parameter ? _i ² (i = 1, 2). The parameters ?_i take arbitrary values in the half-open interval (0, 1]. When the vector parameter ? = (?₁, ?₂) vanishes, the system of elliptic equations degenerates into a system of algebraic equations. When the components ?₁ and (or) ?₂ tend to zero, a double boundary layer with the characteristic width ?₁ and ?₂ appears in the vicinity of the boundary. Using the grid refinement technique and the classical finite difference approximations of the boundary value problem, special difference schemes that converge ?-uniformly at the rate of O(N ^?2ln² N) are constructed, where N = min N _s, N _s + 1 is the number of mesh points on the axis x _s.     

A Singularity of Fertility Decline Is Lot-et-Garonne

Although we generally tend to conceive that fertility decline has been caused by the alterations of socioeconomic conditions, fertility decline in the 19th century in Europe was a reaction-diffusion process independent of them (e.g., see Coale, 1967 Coale , A. J. ( 1967 ). Factors associated with the development of low fertility : An historic summary, in united nations . Proceedings of the World Population Conference, Belgrade , 2 , 205 – 209 . [Google Scholar], 1973 Coale , A. J. ( 1973 ). The demographic transition . Proceedings of the IUSSP International Conference , 1 , 53 – 72 . [Google Scholar]; Becker, 1978 Becker , G. S. ( 1978 ). The economic approach to human behavior . Chicago , IL : The University of Chicago Press . [Google Scholar], 1981 Becker , G. S. ( 1981 ). A treatise on the family . Cambridge , MA : Harvard University Press . [Google Scholar]). Once we postulate a reaction-diffusion process for this phenomenon, we can estimate the velocity of a progressive wave of diffusion. By means of the estimated velocity, we can estimate where a singularity of fertility decline was and when it appeared. The singularity existed in a French district Aquitaine basin. From Lot-et-Garonne in Aquitaine, the reaction-diffusion of fertility decline began to diffuse to all Europe maintaining relative independence of socioeconomic conditions.     

A high order HODIE finite difference scheme for 1D parabolic singularly perturbed reaction-diffusion problems

This paper deals with the numerical approximation of the solution of 1D parabolic singularly perturbed problems of reaction-diffusion type. The numerical method combines the standard implicit Euler method on a uniform mesh to discretize in time and a HODIE compact fourth order finite difference scheme to discretize in space, which is defined on a priori special meshes condensing the grid points in the boundary layer regions. The method is uniformly convergent having first order in time and almost fourth order in space. The analysis of the uniform convergence is made in two steps, splitting the contribution to the error from the time and the space discretization. Although this idea has been previously used to prove the uniform convergence for parabolic singularly perturbed problems, here the proof is based on a new study of the asymptotic behavior of the exact solution of the semidiscrete problems obtained after the time discretization by using the Euler method. Some numerical results are given corroborating in practice the theoretical results.     

随机时滞反应扩散广义细胞神经网络的均值指数稳定性

研究了一类反应扩散广义时滞细胞神经网络在噪声干扰下的指数稳定性.利用Ito公式,Holder不等式,M矩阵性质和微分不等式技巧,给出了系统均值指数稳定的充分条件,并且判断方法简单易操作.最后给出了主要定理的两个应用实例,表明结论的有效性.     

Symmetry Analysis and Conservation Laws to the (2+1)-Dimensional Coupled Nonlinear Extension of the Reaction-Diffusion Equation

In this paper, a detailed Lie symmetry analysis of the (2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple direct method, which is equivalent to Lie point symmetry group actually. Similarity reduction and some exact solutions of the original equation are obtained based on the optimal system of one-dimensional subalgebras. In addition, conservation laws are constructed by employing the new conservation theorem.     

Dynamics of Front Driven by Convection Field in Bistable Media

The front dynamics driven by a convection field in a model of FitzHugh-Nagumo type is studied both analytieMly and numerically. Saddle-node bifurcation induced by the convection field is found by using a singular perturbation analysis of front solutions. Convection field accelerates the B1och front propagating opposite the direction of convection field, but inhibits the Bloch front propagating along the direction of convection field. In addition convection field drives Ising front to travel opposite the direction of convection field.     

Analytical Approach for Nonlinear Partial Differential Equations of Fractional Order

The purpose of the paper is to present analytical and numerical solutions of a degenerate parabolic equation with time-fractional derivatives arising in the spatial diffusion of biological populations. The homotopy-perturbation method is employed for solving this class of equations, and the time-fractional derivatives are described in the sense of Caputo. Comparisons are made with those derived by Adomian's decomposition method, revealing that the homotopy perturbation method is more accurate and convenient than the Adomian's decomposition method. Furthermore, the results reveal that the approximate solution continuously depends on the time-fractional derivative and the proposed method incorporating the Caputo derivatives is a powerful and efficient technique for solving the fractional differential equations without requiring linearization or restrictive assumptions. The basis ideas presented in the paper can be further applied to solve other similar fractional partial differential equations.