Phase Portrait

Col

![[Pasted image 20250905173724.png|${\lambda }_1,{\lambda }_2<0⟹$Stable Nodal Sink, Asymptotically Stable]]

![[Pasted image 20250905173930.png|${\lambda }_1,{\lambda }_2>0⟹$ Nodal Source, Unstable]]

![[Pasted image 20250905180026.png|${\lambda }_1>0{\lambda }_2<0⟹$ Saddle, Unstable]]

Col

![[Pasted image 20250912160920.png|${\lambda }_2<0$, Attractive line of equilibirum, strictly stable|225]]

![[Pasted image 20250912160948.png|${\lambda }_2>0$, Repulsive line of equilibrium, unstable|200]]

Off-Eigenspace Dynamics

For 2D phase portraits, if the eigenvalue signs match, the curves off of the Eigenspaces are parabolas, else hyperbolas.

N-Dimensional Homogenous System Identification

N-Dimensional Homogenous System Identification

e.g. for ${\vec{x}}^{\prime }=A\vec{x}$, where Eigenstuff comes from matrix $A$

$$\begin{array}{rl}{\vec{x}}^{\prime }=A\vec{x}& ⟹\vec{x}=\vec{0}\text{ is equilbrium}\\ (\exists \mathrm{Re}\lambda >0)\vee & \\ (\forall \lambda \mathrm{Re}\lambda \le 0\wedge & \\ \exists \mathrm{Re}\lambda =0\wedge & \\ \forall \mathrm{Re}\lambda =0,\lambda \equiv \neg \text{Defective})& ⟹\text{Unstable}\\ \forall \mathrm{Re}\lambda <0& ⟹\text{Asymptotically Stable}\\ \text{Else}& ⟹\text{Stable}\end{array}$$

See Defective

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