奇点分类 (III)

数学建模

$\Delta = 0$ A, 有两个相同的特征值

$\mathbf{P} \xi \xi (1 \ 2)$ $\mathbf{P}^{-1} \mathbf{A} \mathbf{P} = \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix}$

\begin{cases} \frac{du}{dt} = \lambda u \\ \frac{dv}{dt} = \lambda v \end{cases} \quad \begin{cases} u = u_0 e^{\lambda t} \\ v = v_0 e^{\lambda t} \end{cases} \quad \frac{u}{v} = \frac{u_0}{v_0}

$\alpha < 0, \lambda < 0$
$\alpha > 0, \lambda > 0$

$\frac{d\mathbf{u}}{dt} = \mathbf{P}^{-1} \mathbf{A} \mathbf{P} \mathbf{u}$ $\alpha = \text{tr}(\mathbf{A}) = \lambda_1 + \lambda_2$ $\beta = |\mathbf{A}| = \lambda_1 \lambda_2$

Phase portrait showing a star node (proper node) at the origin. Trajectories are straight lines radiating from the origin. Arrows point inwards, indicating a stable node (when $\lambda < 0$ ). The axes are labeled $u$ (horizontal) and $v$ (vertical).