The Hopf bifurcation from spike solutions for the classical Gierer–Meinhardt system in a onedimensional interval is considered. The existence of time-periodic solution near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of the limit cycle are determined, and it is shown that the limit cycle is unstable.

中文翻译：

---

弱耦合 Gierer-Meinhardt 系统尖峰解的 Hopf 分岔

考虑了一维区间内经典 Gierer-Meinhardt 系统尖峰解的 Hopf 分岔。经典的 Crandall-Rabinowitz 理论严格证明了边界尖峰的 Hopf 分岔参数附近存在时间周期解。确定了极限环稳定性的判据，证明极限环为不稳定.