critical points of plane autonomous system 2

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\begin{cases}\frac{d x}{d t}=\left(-\frac14-x^2\right) y\\\frac{d y}{d t}=x-y\end{cases}

用Matlab程式pplane8绘图, 坐标轴是$x$和$y$ 蓝实线是orbit(即trajectory) 红虚线是nullcline 红点是equilibrium point(即critical point)

在 critical point 附近线性近似:
点 (0,0), $λ_1=λ_2=-1/2$, Jacobian为 $M=\pmatrix{0 & -\frac14 \\1 & -1}$,
特征值相等, 但 $M≠λI$. 它是一个 degenerate sink (在DE1notes22-11-09.pdf中,称为stable inflected node).
特征向量: $v=\pmatrix{1\\2}$.
当 $|x|$ 很大时, $\pmatrix{x\\y}∼c_1te^{-t/2}\pmatrix{1\\2}$.

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关于 stability diagram 见 en.wikipedia.org/wiki/Autonomous_system_(mathematics)
egwald.ca/linearalgebra/lineardifferentialequationsstabilityanalysis.php
Chapter 11 Plane Autonomous Systems

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用TikZ画一个简图