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Last edited by hbghlyj at 2023-5-6 11:55:00\begin{cases}\frac{d x}{d t}=\left(\frac12-x^2\right) y\\\frac{d y}{d t}=x-y\end{cases}
pplane8.m
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用Matlab程式pplane8绘图, 坐标轴是$x$和$y$
蓝实线是orbit, 红虚线是nullcline
红点是equilibrium point(即critical point) |
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在 critical points 附近线性近似:
- 点 (0,0), Jacobian为 $M=\pmatrix{0&\frac12\\1&-1},λ_1=\frac{-1-\sqrt3}2,λ_2=\frac{-1+\sqrt3}2$
$λ_1<0<λ_2$. 它是一个saddle.
特征向量: $v_1=\pmatrix{\frac{1-\sqrt3}2\\1},v_2=\pmatrix{\frac{1+\sqrt3}2\\1}$. - 点 $\left(\frac1{\sqrt2},\frac1{\sqrt2}\right)$ 和 $\left(-\frac1{\sqrt2},-\frac1{\sqrt2}\right)$, Jacobian为 $M=\pmatrix{-1 & 0 \\1 & -1}$, $λ_1=λ_2=-1$,
特征值相等, 但 $M≠λI$. 它是一个 degenerate sink (在DE1notes22-11-09.pdf中,称为inflected node).
特征向量: $\pmatrix{0\\1}$.
关于 stability diagram 见 en.wikipedia.org/wiki/Autonomous_system_(mathematics)
Chapter 11 Plane Autonomous Systems |
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