改进的Leslie--Gower模型中的一个精确结构恒等式及其对范式的影响
An exact structural identity and its normal form consequences in a modified Leslie--Gower model
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中文总结 AI 辅助
该研究针对改进的Leslie--Gower捕食者-猎物模型，证明了其Bogdanov--Takens奇点相关非线性系数间存在精确结构恒等式，可简化高阶退化的搜索并明确表征退化轨迹。
中文摘要 AI 辅助
Zhao与Zhao（2026）通过递归范式计算，在改进的Leslie--Gower捕食者-猎物模型中建立了一个余维四的Bogdanov--Takens奇点。我们证明，控制约化系统限制在该奇点的特殊本征方向上的非线性系数并非相互独立，而是存在一个已在变换后的方程中出现的单一精确有理恒等式，该恒等式生成了所有纯泰勒系数，并证明这些系数同时消失是一个全阶精确性质，而非有限阶巧合。我们进一步表明，该结构恒等式会通过Kuznetsov（2005）的平面Bogdanov--Takens范式算法传播。尽管纯系数恒等消失，但对应的范式系数由混合二次相互作用生成，这使得退化Bogdanov--Takens轨迹得到明确表征，并将高阶Bogdanov--Takens退化的搜索简化为单一明确的代数条件。
英文摘要
Zhao and Zhao (2026) established a codimension-four Bogdanov--Takens singularity in a modified Leslie--Gower predator--prey model through a recursive normal form computation. We show that the nonlinear coefficients governing the restriction of the reduced system to the singularity's distinguished eigendirection are not independent. Instead, a single exact rational identity, already present in the transformed equations, generates the entire family of pure Taylor coefficients and proves that their simultaneous vanishing is an exact all orders property rather than a finite-order coincidence. We further show that this structural identity propagates through the planar Bogdanov--Takens normal form algorithm of Kuznetsov (2005). Although the pure coefficients vanish identically, the corresponding normal form coefficients are generated by mixed quadratic interactions. This leads to an explicit characterization of the degenerate Bogdanov--Takens locus and reduces the search for higher-order Bogdanov--Takens degeneracies to a single explicit algebraic condition.
