双层耦合非对称反应扩散系统中的超点阵斑图

通过线性耦合Brusselator模型和Lengyel-Epstein模型,数值研究了双层耦合非对称反应扩散系统中图灵模之间的相互作用以及斑图的形成机理.模拟结果表明,合适的波数比以及相同的对称性是两个图灵模之间达到空间共振的必要条件,而耦合强度则直接影响了图灵斑图的振幅大小.为了保证对称性相同,两个图灵模的本征值高度要位于一定的范围内.只有失稳模为长波模时,才能对另一个图灵模产生调制作用,并形成多尺度时空斑图.随着波数比的增加,短波模子系统依次经历黑眼斑图、白眼斑图以及时序振荡六边形斑图的转变.研究表明失稳图灵模与处于短波不稳定区域的高阶谐波模之间的共振是产生时序振荡六边形的主要原因.     

反应扩散模型在图灵斑图中的应用及数值模拟

反应扩散方程模型常被用于描述生物学中斑图的形成.从反应扩散模型出发,理论推导得到GiererMeinhardt模型的斑图形成机理,解释了非线性常微分方程系统的稳定常数平衡态在加入扩散项后会发生失稳并产生图灵斑图的过程.通过计算该模型,得到图灵斑图产生的参数条件.数值方法中采用一类有效的高精度数值格式,即在空间离散条件下采用Chebyshev谱配置方法,在时间离散条件下采用紧致隐积分因子方法.该方法结合了谱方法和紧致隐积分因子方法的优点,具有精度高、稳定性好、存储量小等优点.数值模拟表明,在其他条件一定的情况下,系统控制参数κ取不同值对于斑图的产生具有重要的影响,数值结果验证了理论结果.     

空间周期性驱动对双层耦合反应扩散系统中图灵斑图的影响

周期性驱动是控制斑图最有效的方式之一,因此一直是斑图动力学研究的一大热点.自然界中的斑图形成系统大多是多层耦合的非线性系统,周期性驱动对这些多层耦合系统的作用机理人们还不甚了解.本文通过耦合Brusselator (Bru)系统和Lengyel-Epstein (LE)系统,并给LE系统施加一个空间周期性驱动来研究外部驱动对多层耦合系统中图灵斑图的影响.研究发现,只要外部驱动与Bru系统的超临界图灵模(内部驱动模)两者中的一个为长波模时,就可以将LE系统中的次临界图灵模激发,3个模式共同作用从而形成具有3个空间尺度的复杂斑图.若外部驱动和内部驱动模均为短波模,则无法激发此系统的本征次临界图灵模,但满足空间共振时也可以产生超点阵斑图.若LE系统的本征模为超临界图灵模,其自发形成的六边形斑图只有在外部驱动强度较大的情况下才能够产生响应,且其空间对称性受到外部驱动波数的影响.     

双层耦合非对称反应扩散系统中的振荡图灵斑图

采用线性耦合Brusselator模型和Lengyel-Epstein模型,数值研究了双层耦合非对称反应扩散系统中振荡图灵斑图的动力学,并分析了图灵模、高阶模以及霍普夫模之间的相互作用及其对振荡图灵斑图的影响.模拟结果表明,在Lengyel-Epstein模型激发的超临界图灵模k_1的激励下,Brusselator模型中处于霍普夫区域的高阶模3~(1/2)k_1被激发,这两个模式相互作用从而产生了同步振荡六边形斑图.随着控制参数b的增加,该振荡六边形斑图首先经历倍周期分岔进入双倍振荡周期,经历多倍振荡周期后,在霍普夫模式的参与下,最终进入时空混沌态.同步振荡六边形斑图形成的条件是Brusselator模型中的次临界图灵模k_2的本征值高度低于处于霍普夫区域的高阶图灵模3~(1/2)k_1的本征值高度,且两个图灵模之间不存在空间共振关系.当两个图灵模满足空间共振时,系统优先选择空间共振模式,从而产生超点阵斑图.霍普夫模和图灵模共同作用下只能产生非同步振荡图灵斑图.此外,耦合强度对振荡图灵斑图也有重要的影响.     

六边形格子态斑图的数值模拟

采用双层耦合的Lengel-Epstein模型, 通过改变两子系统图灵模的强度比, 获得了四种的六边形格子态和多种非格子态结构. 模拟结果表明: 反应扩散系统的格子态结构由三套子结构叠加而成, 是两图灵模的波数比和强度比共同作用的结果, 两模的强度比决定了三波共振的具体模式; 另外, 系统选择格子态斑图所需的两图灵模的强度比大于非格子态斑图的强度比; 逐步增加两图灵模强度比, 出现的斑图趋于从复杂到简单变化. 深入研究发现: 不同互质数对(a, b)对应的格子态斑图的稳定性不同, 其中(3, 2)对应的格子态结构最为稳定.     

双层耦合Lengel-Epstein模型中的超点阵斑图

用双层耦合的Lengel-Epstein模型, 研究了两个子系统的图灵模对斑图的影响,发现其波数比在斑图的形成和选择过程中起着重要作用.当波数比为1时,双层系统未能发生耦合,只能出现条纹和六边形斑图;当波数比处于1-√17 的范围时,两子系统发生耦合,图灵模之间发生共振相互作用,得到种类丰富的超点阵斑图,包括暗点、点-棒和复杂超六边、Ⅰ-型和Ⅱ-型白眼、类蜂窝和环状超六边等斑图;当波数比大于√17 , 系统选择的斑图类型不再变化,均为环状超六边斑图.数值模拟得到的条纹、六边形、超六边点阵、Ⅱ-型白眼斑图和类蜂窝斑图均已在介质阻挡放电系统实验中观察到. 另外,还得到了超点阵斑图的波数随两个扩散系数乘积D_uD_v的变化曲线,发现其随的D_uD_v增大而减小. 关键词： 耦合系统 超点阵 波数比 数值模拟     

反应扩散系统中时空斑图研究

时空斑图广泛地存在于反应扩散系统中,在延展的布鲁塞尔振子模型中,一维的时空斑图已经被研究过.本文中,我们对布鲁塞尔振子模型进行线性稳定性分析,模拟出两维的时空斑图,进一步阐明斑图形成的机制,形成斑图的机制是由于霍普夫失稳、短波失稳和图灵失稳以及它们之间的相互作用.当系统处于非平衡状态下,布鲁塞尔振子模型可以得到有序的时空斑图.?     

调制驻波光场中二能级原子动量扩散的动力学研究

分析了二能级原子在振幅调制驻波光场作用下动量扩散模型.这个量子系统在经典极限下表现混沌行为.在相同参数条件下,这个系统具有动力学局域特征. 关键词：     

反常扩散与分数阶对流-扩散方程

反常扩散现象在自然界和社会系统中广泛存在.考虑了扩散过程的时间相关和时空相关性，用非局域性的处理方法，在传统的二阶对流 扩散方程基础上，得到了分数阶对流 扩散方程，以此方程来描述反常扩散.在此方程中，弥散项和对时间的导数为分数阶导数所代替.由此分数阶对流 扩散方程，对传统的费克扩散定律进行推广，得到了广义的分数费克扩散定律，分数费克扩散定律说明某时刻空间中某点的流量不仅与其领域内的浓度梯度有关，而且与整个空间中其他不同点的粒子浓度、浓度变化的历史，甚至初始时刻的浓度有关.讨论了方程的解——分数稳定分布，并由此说明了扩散运动的平均平方位移是运移时间的非线性函数. 关键词： 扩散 分数阶微积分 稳定分布（Lévy分布） 费克扩散定律     

低杂波电流驱动中径向扩散效应的数值模拟

应用改进后的程序详细计算了不同径向扩散系数对低杂波电流驱动剖面分布的影响。通过计算发现:考虑径向扩散效应后,驱动电流分布变平展宽,电流驱动的分布随着扩散系数的增大逐渐向外层移动,由局域性分布演化成非局域性分布;驱动电流的大小和效率随着扩散系数的增大而降低。     

Numerical simulation and analysis of complex patterns in a two-layer coupled reaction diffusion system

The resonance interaction between two modes is investigated using a two-layer coupled Brusselator model. When two different wavelength modes satisfy resonance conditions, new modes will appear, and a variety of superlattice patterns can be obtained in a short wavelength mode subsystem. We find that even though the wavenumbers of two Turing modes are fixed, the parameter changes have influences on wave intensity and pattern selection. When a hexagon pattern occurs in the short wavelength mode layer and a stripe pattern appears in the long wavelength mode layer, the Hopf instability may happen in a nonlinearly coupled model, and twinkling-eye hexagon and travelling hexagon patterns will be obtained. The symmetries of patterns resulting from the coupled modes may be different from those of their parents, such as the cluster hexagon pattern and square pattern. With the increase of perturbation and coupling intensity, the nonlinear system will convert between a static pattern and a dynamic pattern when the Turing instability and Hopf instability happen in the nonlinear system. Besides the wavenumber ratio and intensity ratio of the two different wavelength Turing modes, perturbation and coupling intensity play an important role in the pattern formation and selection. According to the simulation results, we find that two modes with different symmetries can also be in the spatial resonance under certain conditions, and complex patterns appear in the two-layer coupled reaction diffusion systems.     

Uphill anomalous transport in a deterministic system with speed-dependent friction coefficient

We investigate the transport of a deterministic Brownian particle theoretically, which moves in simple onedimensional, symmetric periodic potentials under the influence of both a time periodic and a static biasing force. The physical system employed contains a friction coefficient that is speed-dependent. Within the tailored parameter regime, the absolute negative mobility, in which a particle can travel in the direction opposite to a constant applied force, is observed.This behavior is robust and can be maximized at two regimes upon variation of the characteristic factor of friction coefficient. Further analysis reveals that this uphill motion is subdiffusion in terms of localization(diffusion coefficient with the form D(t) ~t~(-1) at long times). We also have observed the non-trivially anomalous subdiffusion which is significantly deviated from the localization; whereas most of the downhill motion evolves chaotically, with the normal diffusion.     

Turing bifurcation in a reaction-diffusion system with density-dependent dispersal

Motivated by the recent finding [N. Kumar, G.M. Viswanathan, V.M. Kenkre, Physica A 388 (2009) 3687] that the dynamics of particles undergoing density-dependent nonlinear diffusion shows sub-diffusive behaviour, we study the Turing bifurcation in a two-variable system with this kind of dispersal. We perform a linear stability analysis of the uniform steady state to find the conditions for the Turing bifurcation and compare it with the standard Turing condition in a reaction-diffusion system, where dispersal is described by simple Fickian diffusion. While activator-inhibitor kinetics are a necessary condition for the Turing instability as in standard two-variable systems, the instability can occur even if the diffusion constant of the inhibitor is equal to or smaller than that of the activator. We apply these results to two model systems, the Brusselator and the Gierer-Meinhardt model.     

An efficient and robust numerical algorithm for estimating parameters in Turing systems

We present a new algorithm for estimating parameters in reaction–diffusion systems that display pattern formation via the mechanism of diffusion-driven instability. A Modified Discrete Optimal Control Algorithm (MDOCA) is illustrated with the Schnakenberg and Gierer–Meinhardt reaction–diffusion systems using PDE constrained optimization techniques. The MDOCA algorithm is a modification of a standard variable step gradient algorithm that yields a huge saving in computational cost. The results of numerical experiments demonstrate that the algorithm accurately estimated key parameters associated with stationary target functions generated from the models themselves. Furthermore, the robustness of the algorithm was verified by performing experiments with target functions perturbed with various levels of additive noise. The MDOCA algorithm could have important applications in the mathematical modeling of realistic Turing systems when experimental data are available.     

Random walks on the Comb model and its generalizations

Microscopic models with anomalous diffusion, which include the Comb model and its generalization for the finite width of the backbone, have been considered in this paper. The physical mechanisms of the subdiffusion random walks have been established. The first comes from the permanent return of the diffusing particle to the initial point of the diffusion due to "effective reducing" of the dimensionality of the considered system to the quasi-one-dimensional system. This physical mechanism has been obtained in the Comb model and in the model with a strip. The second mechanism of the subdiffusion is connected with random capture on the traps of diffusing particles and their ensuing random release from the traps. It has been shown that these different mechanisms of subdiffusion have been described by the different generalized diffusion equations of fractional order. The solutions of these different equations have been obtained, and the physical sense of the fractional order generalized equations has been discussed.     

双层非线性耦合反应扩散系统中复杂Turing斑图

采用双层耦合的Brusselator模型, 研究了两个子系统非线性耦合时Turing 模对斑图的影响, 发现两子系统Turing 模的波数比和耦合系数的大小对斑图的形成起着重要作用. 模拟结果表明: 斑图类型随波数比值的增加, 从简单斑图发展到复杂斑图; 非线性耦合项系数在0–0.1时, 系统1中短波模在系统2失稳模的影响下不仅可形成简单六边形、四边形和条纹斑图, 两模共振耦合还可以形成蜂窝六边形、超六边形和复杂的黑眼斑图等超点阵图形, 首次在一定范围内调整控制参量观察到由简单正四边形向超六边形斑图的转化过程; 耦合系数在0.1–1时, 系统1中短波模与系统2失稳模未发生共振耦合仅观察到与系统2相同形状的简单六边形、四边形和条纹斑图. 关键词： Brusselator模型 非线性耦合 Turing模     

Spatial Pattern of an Epidemic Model with Cross-diffusion

Pattern formation of a spatial epidemic model with both self- and cross-diffusion is investigated. From the Turing theory, it is well known that Turing pattern formation cannot occur for the equal self-diffusion coefficients. However, combined with cross-diffusion, the system will show emergence of isolated groups, i.e., stripe-like or spotted or coexistence of both, which we show by both mathematical analysis and numerical simulations. Our study shows that the interaction of self- and cross-diffusion can be considered as an important mechanism for the appearance of complex spatiotemporal dynamics in epidemic models.