A flower graph consists of a half line and $N$ symmetric loops connected at a single vertex with $N \geq 2$ (it is called the tadpole graph if $N = 1$). We consider positive single-lobe states on the flower graph in the framework of the cubic nonlinear Schrodinger equation. The main novelty of our paper is a rigorous application of the period function for second-order differential equations towards understanding the symmetries and bifurcations of standing waves on metric graphs. We show that the positive single-lobe symmetric state (which is the ground state of energy for small fixed mass) undergoes exactly one bifurcation for larger mass, at which point $(N-1)$ branches of other positive single-lobe states appear: each branch has $K$ larger components and $(N-K)$ smaller components, where $1 \leq K \leq N-1$. We show that only the branch with $K = 1$ represents a local minimizer of energy for large fixed mass, however, the ground state of energy is not attained for large fixed mass. Analytical results obtained from the period function are illustrated numerically.

中文翻译：

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花图上的驻波

一个花图由一条半线和 $N$ 对称环组成，这些环在单个顶点处与 $N \geq 2$ 相连（如果 $N = 1$，则称为蝌蚪图）。我们在三次非线性薛定谔方程的框架内考虑花图上的正单瓣状态。我们论文的主要新颖之处在于严格应用二阶微分方程的周期函数来理解度量图上驻波的对称性和分岔。我们表明正单瓣对称状态（它是小固定质量的能量基态）对于较大质量恰好经历一个分叉，此时出现其他正单瓣状态的 $(N-1)$ 分支：每个分支有 $K$ 个较大的组件和 $(NK)$ 个较小的组件，其中 $1 \leq K \leq N-1$。我们表明只有 $K = 1$ 的分支代表大固定质量的局部能量极小值，但是，大固定质量没有达到能量的基态。从周期函数获得的分析结果以数字方式说明。