The main goal of this paper is to understand the formation of hexagonal patterns from the dynamical transition theory point of view. We consider the transitions from a steady state of an abstract nonlinear dissipative system. To shed light on the formation of mixed mode patterns such as the hexagonal pattern, we consider the case where the linearized operator of the system has two critical real eigenvalues, at a critical value λ_c of a control parameter λ with associated eigenmodes having a roll and rectangular pattern. By using center manifold reduction, we obtain the reduced equations of the system near the critical transition value λ_c. By a through analysis of these equations, we fully characterize all possible transition scenarios when the coefficients of the quadratic part of the reduced equations do not vanish. We consider three problems, two variants of the 2D Swift-Hohenberg equation and the 3D surface tension driven convection, to demonstrate that all the main theoretical results we obtain here are indeed realizable.

本文的主要目的是从动力学过渡理论的角度了解六角形图案的形成。我们考虑从抽象非线性耗散系统的稳态转变。棚上的混合模式的图案形成的光，例如六边形图案，我们考虑其中系统的线性算子具有两个关键的实特征值，在一个临界值的情况下λ _Ç的控制参数的λ与具有辊相关联的本征模式和矩形图案。通过使用中心歧管减少，我们获得了系统的临界转变值附近的低方程λ _Ç。通过对这些方程的透彻分析，我们可以全面刻画当简化方程的二次部分的系数不消失时所有可能的过渡情况。我们考虑了三个问题，即2D Swift-Hohenberg方程的两个变体和3D表面张力驱动的对流，以证明我们在此获得的所有主要理论结果确实可以实现。