猎物零倾线上的迹场：可容许分支上平面Hopf分岔的临界性
The trace field on the prey nullcline: criticality of planar Hopf bifurcations on the admissible branch
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中文总结 AI 辅助
本研究聚焦平面捕食-食饵系统食饵零倾线的迹场，重构第一李雅普诺夫系数，推导其闭式表达式与分解式，揭示Hopf分岔的局域化规律及临界性修正的桥梁恒等式，并通过典型模型验证。
中文摘要 AI 辅助
我们研究平面捕食-食饵系统食饵零倾线上的迹场，围绕这一局部几何结构重构第一李雅普诺夫系数。对于图像型第一零倾线，我们证明$J_{11}=Pg'$，其中$P=-f_{1y}$为边际捕食率。在捕食者自阻尼条件下，该恒等式将Hopf分岔定位于食饵零倾线的上升分支。利用适配零倾线的坐标与阿达马引理，我们推导得到第一李雅普诺夫系数$\ell_1$的五项闭式表达式，该式对依赖捕食者的功能响应依然成立。我们建立了三种不同的结构约化机制：两种是与临界特征向量的三次配对及零倾线拉直的普遍结果，第三种则为高斯类系统特有。将完整系数与$y$仿射三阶截断式对比，得到精确分解式$\ell_1=\ell_1^{\mathrm{aff}}+Δ$，其中$Δ$由两条依赖捕食者的三次通道及一个正结构因子共同决定。在Hopf轨迹上，我们推导出一个桥梁恒等式，表明定位分岔的同一物理量$J_{11}=Pg'$同时对临界性修正中迹场的切向导数起加权作用。对代表性捕食-食饵模型的精确与高精度验证，复现了上述解析恒等式与数值结果。
英文摘要
Along the prey nullcline $y=g(x)$ of a planar system, the Jacobian entry satisfies $J_{11}=P\,g'$, where $P=-f_{1y}|_γ$. Under $P>0$ and predator self-damping, the Hopf trace condition forces $g'(x^\ast)>0$: critical points of $g$ are spectral barriers, so the Hopf locus lies on ascending branches. This geometric localization determines where oscillatory instability can occur; we determine how the bifurcation unfolds there. Let $D=f_{1x}+f_{2y}$ and $τ=D|_γ$. Using nullcline coordinates $(u=x,v=y-g(x))$ and Hadamard's lemma gives $\dot u=-vH$, $\dot v=W$, with $H(u,0)=P$. The mixed jet of $W$ is determined by the jets of $τ$ and $ν=f_2|_γ$, yielding a closed five-term formula for the first Lyapunov coefficient $\ell_1$ in trace-field form, valid for predator-dependent functional responses. Three distinct reductions of the cubic jet are established. In general planar Hopf bifurcations, eigenvector pairing gives $\partial\ell_1/\partial f_{1yyy}=\partial\ell_1/\partial f_{2xxx}=0$. Straightening the nullcline removes two further slots, while within the Gause class the reparametrization $φ\mapsto g$ removes $p'''$. They have different scopes. For predator-dependent responses, comparison with the $y$-affine truncation of the $3$-jet at fixed linear part gives $\ell_1=\ell_1^{\mathrm{aff}}+Δ$, where $Δ=Λ\bigl(f_{1xyy}+ω_0^{-2}\langle\partial_x f,\nabla D\rangle f_{1yy}\bigr)$ and $Λ>0$. On the Hopf locus, $\langle\partial_x f,\nabla D\rangle=J_{11}τ'+(ω_0^2/P)D_y$. Thus the same quantity $J_{11}=Pg'$ that localizes the bifurcation at first order reappears at third order as the weight of the tangential trace derivative. The bridge identity is independent of eigenvector normalization, while $\ell_1,\ell_1^{\mathrm{aff}},Δ,Λ$ scale by $|c|^2$ under $q\mapsto cq$; their signs and ratios are unchanged.
发表机构
- División Académica de Ciencias Básicas, Universidad Juárez Autónoma de Tabasco（塔巴斯科大学自治朱阿雷斯学院基础科学分院）
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