由平面常微分方程组定义的可持续供应链中的Hopf分岔与周期解
Hopf bifurcation and periodic solutions in a sustainable supply chain defined by a planar system of ordinary differential equations
浏览论文内容
中文总结 AI 辅助
本研究针对描述可持续供应链产品-资源相互作用的平面无时滞常微分方程组，分析其平衡点稳定性，揭示环境容量超阈值时的超临界Hopf分岔现象，推导临界阈值并经数值模拟验证，得到周期解的相关估计。
中文摘要 AI 辅助
我们分析了描述可持续供应链中产品-资源相互作用的平面常微分方程组的平衡点稳定性与分岔问题。尽管供应链模型中的周期行为常出现在高维系统或时滞系统中，但本研究显示在平面无时滞系统中也会出现Hopf分岔。我们证明，当环境容量超过临界阈值时，内部平衡点会通过超临界Hopf分岔失去稳定性，该阈值取决于最大生产率、生产率达到一半最大值时的资源水平、需求率及再制造率。我们明确推导了该阈值，并借助所得的Hopf标准型给出了Hopf分岔附近周期解的振幅与周期估计，随后通过数值模拟进一步验证了分析结果。
英文摘要
We analyze the stability of the equilibria and bifurcations of a planar system of ordinary differential equations describing the product-resource interaction in a sustainable supply chain. While periodic behavior in supply chain models has often been documented either in systems of higher dimensions or in delayed systems, here a Hopf bifurcation arises in a planar and delay-free system. We show that the interior equilibrium loses stability through a supercritical Hopf bifurcation as the environmental capacity exceeds a critical threshold that depends on the maximum production rate, the resource level at which production reaches half its maximum rate, and the demand and remanufacturing rates. We derive this threshold explicitly and provide estimates for the amplitude and period of the periodic solution close to the Hopf bifurcation, by means of the resulting Hopf normal form. We then further substantiate our analytical results through numerical simulations.
