论文标题
反应扩散系统中的分叉分析和稳态模式，并随着自我和交叉扩张而增强
Bifurcation analysis and steady state patterns in reaction-diffusion systems augmented with self- and cross-diffusion
论文作者
论文摘要
在本文中，我们将使用增强的灰色 - 斯科特系统作为一般示例进行了一项研究，对反应扩散系统的长期行为进行了增强。该方法仍然是通用的，因此适用于其他系统。时间模型（非线性抛物线系统）的模拟揭示了稳态的存在，通常与能量耗散有关。为了直接评估时间系统的稳态（非线性椭圆系统），并通过其解决方案进行了验证，提供了一种基于混合有限元方法的牛顿方法。使用傅立叶分析的线性稳定性分析（LSA）围绕均匀平衡进行，并在非均匀的频谱分析周围进行。对于后者，光谱问题是数值解决的。报告了多参数分叉。原始的稳态图案已揭幕，仅在线性扩散中可观察到。进行了两个关键的观察：模式与系统初始条件的依赖性，以及对域几何形状的依赖性。
In this article, we carry out a study of long-term behavior of reaction-diffusion systems augmented with self- and cross-diffusion, using an augmented Gray-Scott system as a general example. The methodology remains generic, and is therefore applicable to other systems. Simulations of the temporal model (nonlinear parabolic system) reveal the presence of steady states, often associated with energy dissipation. A Newton method based on a mixed finite element method is provided, in order to directly evaluate the steady states (nonlinear elliptic system) of the temporal system, and is validated against its solutions. Linear stability analysis (LSA) using Fourier analysis is carried out around homogeneous equilibria, and using spectral analysis around non-homogeneous ones. For the latter, the spectral problem is solved numerically. A multi-parameter bifurcation is reported. Original steady state patterns are unveiled, not observable with linear diffusion only. Two key observations are made: a dependency of the pattern with the initial condition of the system, and a dependency on the geometry of the domain.
