AI 中文总结
研究多项式动力系统有限时间爆破问题，通过将相空间紧致化与计算代数技术结合，开发数值框架识别参数空间中爆破区域，用自动计算工具取代特定问题手工计算来分析爆破区域。
AI 中文摘要
有限时间爆破是动力模型变得奇异的一种方式，常表明建模的物理系统或模型本身的崩溃。确定爆破是否发生以及对于哪些参数值和初始条件发生，是分析非线性动力系统的基本问题。我们开发了一个数值框架，用于识别参数空间中由具有多项式右侧的一阶常微分方程组控制的动力系统对于至少一个初始条件表现出有限时间爆破的区域。该方法将相空间紧致化与计算代数技术相结合，产生揭示爆破和非爆破区域的分区参数景观。通过几个例子，我们表明该方法用一个自动计算工具取代了特定问题的手工计算，用于分析参数依赖动力系统中的爆破区域。
英文摘要
Finite-time blow-up is one of the ways in which a dynamical model can become singular, often signaling the breakdown of either the modeled physical system or the model itself. Determining whether blow-up occurs, and for which parameter values and initial conditions, is therefore a fundamental problem in the analysis of nonlinear dynamical systems. We develop a numerical framework for identifying regions of parameter space in which a dynamical system governed by a system of first-order ordinary differential equations with polynomial right-hand sides exhibits finite-time blow-up for at least one initial condition. The approach combines compactification of the phase space with computational algebraic techniques, producing partitioned parameter landscapes that reveal blow-up and non-blow-up regimes. Through several examples, we show that the method replaces problem-specific hand calculations with an automated computational tool for analyzing blow-up regions in parameter-dependent dynamical systems.
