ODE振荡爆破解的代数-动力学对应关系
On Algebraic-Dynamical Correspondence of Oscillatory Blow-Up Solutions for ODEs
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- Institute of Mathematics for Industry, Kyushu University（九州大学产业数値科学研究所）
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中文总结 AI 辅助
本文为自治ODE的I型振荡爆破解建立了代数-动力学对应关系，将平衡律周期解与无穷远周期轨道关联，并给出无需显式紧化的存在性判据，同时阐明了两类稳定性信息的差异。
中文摘要 AI 辅助
我们为具有渐近拟齐次向量场的自治ODE的一类I型振荡爆破解建立了代数-动力学对应关系。由爆破渐近展开导出的平衡律的周期解与去奇异化向量场在无穷远视界上的周期轨道相关联。与平稳情形不同，该对应关系涉及一个正的周期标度函数和一个非平凡的重新参数化时间。我们进一步建立了线性化结构的对应关系：区分相和标度方向生成不变子丛，而其余特征乘子通过相关的商丛来识别。这些结果给出了基于平衡律周期解及其特征乘子的I型周期爆破解存在性的判据，无需显式构造紧化或无穷远处的周期轨道。该理论还阐明了平衡律中的稳定性信息与无穷远处动力学中的稳定性信息之间的差异。两个例子说明了该对应关系。
英文摘要
We develop an algebraic-dynamical correspondence for type-I oscillatory blow-up solutions of autonomous ODEs with asymptotically quasi-homogeneous vector fields. Periodic solutions of the balance law arising from asymptotic expansions of blow-ups are related to periodic orbits on the horizon for the desingularized vector field. Unlike the stationary case, the correspondence involves a positive periodic scaling function and a nontrivial time reparametrization. We further establish the correspondence of linearized structures: distinguished phase and scaling directions generate invariant subbundles, while the remaining characteristic multipliers are identified through associated quotient bundles. These results yield a criterion for the existence of type-I periodic blow-up solutions based on periodic solutions of the balance law and their characteristic multipliers, without explicitly constructing compactifications or periodic orbits at infinity. The theory also clarifies the difference between stability information in the balance law and in dynamics at infinity. Two examples illustrate the correspondence.
