常微分方程定性理论与稳定性理论的哲学思考

Philosophical Reflections on Qualitative Theory and Stability Theory of Ordinary Differential Equations

常微分方程在经历了长期的求精确解的努力后逐渐停滞,庞加莱在分析的基础上引入几何方法,开创了常微分方程定性理论,李雅普诺夫则在庞加莱定性分析的基础上,转而进入了新的稳定性研究。通过对两者的比较研究,我们能够对科学历程中新思想、新理论的产生和发展规律有所感悟。

Ordinary differential equations stagnated gradually after searching hard for the exact solutions of differential equations for a long time. Poincaré used the integrated idea to initiate qualitative theory of differential equations. He also introduced geometric method on the base of analysis and ensured the quality of solutions though the integrated and holistic viewpoint. On the basis of Poincaré's qualitative analysis, Liapunov entered into the new research of stability theory to make the study of solutions more specific and practical. By comparing the qualitative theory and stability theory on differential equations, one can understand the contents and methods of their contents and methods deeply and comprehensively, and can also perceive the laws of the emergence and development of new ideas and theories in the scientific course .