Comparison of linear Turing instability analysis with numerical integrations for the dispersive propagator.

For selected orders and conduction velocities linear Turing instability analyses of Eqns. (66)–(68) were used to predict the critical Turing-Hopf , , and , cf. Fig. 9. For numerical simulations, a somewhat larger than was chosen. The space-averaged 1D temporal Fourier spectrum was used to estimate as the maximum of . The time-averaged 2D spatial Fourier transform was used to obtain two estimates: as the for which is maximal; and as the for which the mean of over a circle around the origin with radius is maximal. For the estimates grid time series of 50 time units with (500 samples total) were recorded, after initial “transients” of 100 () time units were discarded. The spatial grid was () with discretization steps .

针对选定的阶次与传导速度，本文采用方程组(66)–(68)的线性图灵不稳定性（Turing instability）分析，对临界图灵-霍普夫（Turing-Hopf）相关参数进行了预测，参见图9。数值模拟环节中，选取了略大于[某参数]的[某参数]。采用空间平均一维时间傅里叶频谱（Fourier spectrum），以[某量]的最大值估算[目标参数]。通过时间平均二维空间傅里叶变换（Fourier transform），可得到两项估算结果：其一为使[某量]取最大值的[参数]；其二为使以原点为中心、半径为[某值]的圆周上[某量]的平均值取最大值的[参数]。为开展上述估算，我们记录了时长为50个时间单位、采样间隔为[某值]的网格时间序列（总采样数共计500个），并舍弃了初始时长为100（[某单位]）时间单位的暂态过程。空间网格尺寸为[某尺寸]（[某维度参数]），离散步长为[某值]。