Axisymmetric and nonaxisymmetric patterns in the cubic-quintic Swift-Hohenberg equation posed on a disk with Neumann boundary conditions are studied via numerical continuation and bifurcation analysis. Axisymmetric localized solutions in the form of spots and rings known from earlier studies persist and snake in the usual fashion until they begin to interact with the boundary. Depending on parameters, including the disk radius, these states may or may not connect to the branch of domain-filling target states. Secondary instabilities of localized axisymmetric states may create multiarm localized structures that grow and interact with the boundary before broadening into domain-filling states. High azimuthal wave number wall states referred to as daisy states are also found. Secondary bifurcations from these states include localized daisies, i.e., wall states localized in both radius and angle. Depending on parameters, these states may snake much as in the one-dimensional Swift-Hohenberg equation, or invade the interior of the domain, yielding states referred to as worms, or domain-filling stripes.

中文翻译：

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圆盘上三次五次 Swift-Hohenberg 方程中的局部和扩展模式

通过数值延拓和分岔分析，研究了在具有 Neumann 边界条件的圆盘上提出的三次五次 Swift-Hohenberg 方程中的轴对称和非轴对称模式。早期研究中已知的点和环形式的轴对称局部解以通常的方式持续和蛇行，直到它们开始与边界相互作用。根据参数，包括磁盘半径，这些状态可能会或可能不会连接到域填充目标状态的分支。局部轴对称状态的二次不稳定性可能会产生多臂局部结构，在扩展到域填充状态之前，这些结构会生长并与边界相互作用。还发现了称为菊花状态的高方位波数壁状态。来自这些状态的次级分岔包括局部雏菊，即，壁状态集中在半径和角度。根据参数的不同，这些状态可能会像一维 Swift-Hohenberg 方程中那样蜿蜒曲折，或者侵入域的内部，产生称为蠕虫或域填充条纹的状态。