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Analysis of Liapunov's Second Method in Stability of Nonlinear Differential Equations

Explore the drawbacks of determining stability of solutions to nonlinear differential equations through linear approximations. Learn about Liapunov's second method, which involves constructing special functions and analyzing their total derivative signs along trajectory lines to ascertain solution stability. Delve into the general theory and geometric interpretation of Liapunov's second method. Gain insights into using Liapunov functions and the direct series representation of solutions. This study also incorporates the consideration of an undamped mathematical pendulum system. Visualize the explanation of Liapunov's second method from the perspective of positive definite function gradients.

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Analysis of Liapunov's Second Method in Stability of Nonlinear Differential Equations

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  1. Liapunov第二方法

  2. 一、按线性近似判定非线性微分方程解的稳定性的缺陷一、按线性近似判定非线性微分方程解的稳定性的缺陷 线性化系统为： 二、考虑无阻力数学摆

  3. 取函数 性质:

  4. Liapunov第二方法思想：构造特殊函数，通过沿方程的轨线对该函数求全导数的符号来确定方程解的稳定性．Liapunov第二方法思想：构造特殊函数，通过沿方程的轨线对该函数求全导数的符号来确定方程解的稳定性． 特殊函数　　　Liapunov函数　　　　　　　 Liapunov第一方法：直接把解表示成级数形式

  5. 三、 Liapunov第二方法的一般理论

  6. 四、Liapunov第二方法的几何解释

  7. 图1

  8. 从定正函数梯度角度解释Liapunov第二方法 　　图２

  9. 谢 谢 大 家 再见

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