AI 中文总结
该研究针对两类对称非线性分数阶微分方程，通过分析其局部与全局等变分岔，揭示了周期解的存在性及连通解分支的延拓规律。
AI 中文摘要
我们研究两类具有对称性的非线性分数阶微分方程周期解的局部与全局分岔。对于单侧周期Weyl方程，非零时间傅里叶模态会产生复特征函数，自然形成双参数分岔问题，其关联的局部不变量由绕数与扭曲等变度表示；相比之下，周期Riesz方程由实谱量支配，其局部分岔不变量通过等变度在孤立临界值处的跳跃得到。在两种情形中，空间同型分量与时间傅里叶模态的分解决定了临界表示与分岔解的可能对称性。非零局部不变量保证了附近非平凡周期解的存在性，而对应的全局分岔定理则描述了连通解分支从平凡分支的延拓过程。
英文摘要
We investigate local and global bifurcation of periodic solutions for two classes of nonlinear fractional differential equations with symmetry. For the one-sided periodic Weyl equation, the nonzero temporal Fourier modes give rise to complex characteristic functions, leading naturally to a two-parameter bifurcation problem. The associated local invariant is expressed in terms of winding numbers and twisted equivariant degree. In contrast, the periodic Riesz equation is governed by real spectral quantities, and its local bifurcation invariant is obtained from the jump of the equivariant degree across isolated critical values. In both settings, the decomposition into spatial isotypical components and temporal Fourier modes determines the critical representations and the possible symmetries of bifurcating solutions. A nonvanishing local invariant yields the existence of nearby nontrivial periodic solutions, while the corresponding global bifurcation theorems describe the continuation of connected solution components away from the trivial branch.
