论文标题
一般二维域上Schnakenberg反应扩散系统中局部斑点模式的振荡转化不稳定性
Oscillatory translational instabilities of localized spot patterns in the Schnakenberg reaction-diffusion system on general 2-D domains
论文作者
论文摘要
对于有界的二维平面域$ω$，我们研究了域几何形状对$ n $ spot平衡解决方案的振荡转化不稳定性的影响，用于使用$ \ mo（\ eps^2）\ mo（1）$（1）$（1）$（1）$（1）$（1）$（1）$激活的schnakenberg反应 - 反应 - $（1）$（1） $ n $ -Spot平衡的特征是激活器浓度在$ω$中呈指数小的浓度，除了$ n $分离的本地化区域$ \ mo（\ eps）$范围。我们使用匹配的渐近分析的方法来分析HOPF分叉阈值，在该阈值中，平衡对转移扰动变得不稳定，从而导致$ \ mo（\ eps^2）$ - 频率 - 频率 - 频率振荡。我们发现，这些扰动的稳定性受非线性矩阵 - 元素价问题的控制，其特征向量是$ 2N $ - 向量，它表征了振荡的可能模式（方向）。 $ 2N \ times 2n $矩阵包含与$ω$上的某个绿色功能相关的术语，该功能编码了几何效果。对于具有radius的特殊情况，极性坐标为$ r = 1 +σf（θ）$，带有\ red {$ 0 <\ varepsilon \ llσ\ ll 1 $}，$θ\ in [0,2π）$ in [0,2π）$和$ f（θ）$ f（θ）$ $ $2π$ - periers $ coeffiers，我们仅显示$ co $ - 我们仅显示$ coeftiers $ coeftiers $ coeftiels $ coefterial $ f $影响$σ$的领先订单分叉阈值。我们进一步表明，当$ f（θ）= \cos2θ$时，振荡的主要振荡模式朝着平行于扰动磁盘的较长轴的方向。针对各种域$ω$和$ n $ spot Equilibria进行了全面Schnakenberg PDE的数值研究，以确认渐近结果，并证明域几何形状如何影响阈值和主要的振荡模式。
For a bounded 2-D planar domain $Ω$, we investigate the impact of domain geometry on oscillatory translational instabilities of $N$-spot equilibrium solutions for a singularly perturbed Schnakenberg reaction-diffusion system with $\mO(\eps^2) \ll \mO(1)$ activator-inhibitor diffusivity ratio. An $N$-spot equilibrium is characterized by an activator concentration that is exponentially small everywhere in $Ω$ except in $N$ well-separated localized regions of $\mO(\eps)$ extent. We use the method of matched asymptotic analysis to analyze Hopf bifurcation thresholds above which the equilibrium becomes unstable to translational perturbations, which result in $\mO(\eps^2)$-frequency oscillations in the locations of the spots. We find that stability to these perturbations is governed by a nonlinear matrix-eigenvalue problem, the eigenvector of which is a $2N$-vector that characterizes the possible modes (directions) of oscillation. The $2N\times 2N$ matrix contains terms associated with a certain Green's function on $Ω$, which encodes geometric effects. For the special case of a perturbed disk with radius in polar coordinates $r = 1 + σf(θ)$ with \red{$0< \varepsilon \ll σ\ll 1$}, $θ\in [0,2π)$, and $f(θ)$ $2π$-periodic, we show that only the mode-$2$ coefficients of the Fourier series of $f$ impact the bifurcation threshold at leading order in $σ$. We further show that when $f(θ) = \cos2θ$, the dominant mode of oscillation is in the direction parallel to the longer axis of the perturbed disk. Numerical investigations on the full Schnakenberg PDE are performed for various domains $Ω$ and $N$-spot equilibria to confirm asymptotic results and also to demonstrate how domain geometry impacts thresholds and dominant oscillation modes.
