无穷远处动力学中具有法向双曲性质的 ODE 的爆破行为
Blow-up behavior for ODEs with normally hyperbolic nature in dynamics at infinity
AI总结：
本文通过无穷远处动力学描述 ODE 的爆破行为，利用相空间闭嵌入和时间尺度变换将爆破描述转化为地平线上紧轨迹的跟踪性质，并证明具有渐近相位的双曲不变流形可诱导爆破。
摘要：
我们通过自治系统以及非自治系统中具有复杂渐近行为的无穷远处动力学来描述 ODE 的爆破行为。基于先前的研究，相空间的闭嵌入变体以及由无穷远处向量场结构决定的时间尺度变换，将我们对爆破的描述转化为解开地平线上（表达无穷远的几何对象）的（预）紧轨迹的跟踪性质，并具有特定的收敛率。在几何上，该描述由地平线上不变集的渐近相位组织。非自治系统中的爆破解可以用类似的方式描述。作为推论，地平线上具有渐近相位的法向或部分双曲不变流形被证明可以诱导爆破。
英文摘要：
We describe blow-up behavior for ODEs by means of dynamics at infinity with complex asymptotic behavior in autonomous systems, as well as in nonautonomous systems. Based on preceding studies, a variant of closed embeddings of phase spaces and the time-scale transformation determined by the structure of vector fields at infinity reduce our description of blow-ups to unravel the shadowing property of (pre)compact trajectories on the horizon, the geometric object expressing the infinity, with the specific convergence rates. Geometrically, this description is organized by asymptotic phase of invariant sets on the horizon. Blow-up solutions in nonautonomous systems can be described in a similar way. As a corollary, normally, or partially hyperbolic invariant manifolds on the horizon possessing asymptotic phase are shown to induce blow-ups.
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