AI 中文总结
研究四维哈密顿系统周期轨道分支处局部弗洛尔复形链级变化，借助自治系统弗洛尔柱面计数定理及构造特定模型哈密顿量，将计数简化，明确确定了七种突变。
AI 中文摘要
我们计算了四维哈密顿系统中周期轨道一般分支处局部弗洛尔复形的链级变化。尽管局部弗洛尔同调在分支过程中是不变的，但基础链复形会发生突变：周期轨道及其康利 - 曾德指数会不连续变化，弗洛尔微分也必须改变以进行补偿。有两个工具使这种突变可计算。一是证明了自治系统多重覆盖轨道之间弗洛尔柱面的弗洛尔级联计数定理；二是针对迈耶分类中的每种分支类型以及另外两种\(\mathbb{Z}_2\)对称类型，构造了一个模型哈密顿量，其弗洛尔柱面是平面梯度轨迹的提升，即梯度旋转，将计数简化为有限维莫尔斯理论和级联计数。我们明确确定了所有七种突变。
英文摘要
We compute the chain-level change of local Floer complexes across the generic bifurcations of periodic orbits in four-dimensional Hamiltonian systems. Although local Floer homology is invariant across a bifurcation, the underlying chain complex mutates: periodic orbits and their Conley--Zehnder indices change discontinuously, and the Floer differential must change to compensate. Two tools make this mutation computable. First, we prove a Floer cascade counting theorem for Floer cylinders between multiply covered orbits of an autonomous system, which is not specific to the bifurcation analysis carried out here. Second, for each bifurcation type in Meyer's classification, and for two additional $\mathbb{Z}_2$-symmetric types, we construct a model Hamiltonian whose Floer cylinders are lifts of planar gradient trajectories, which we call the gradient revolutions, reducing the count to finite-dimensional Morse theory and cascade counting. We determine all seven mutations explicitly.
