Stability

Stable

An equilibrium point is said to be stable if for some initial value close to the equilibrium point, the solution will eventually stay close to the equilibrium point And vice versa

Asymptotically Stable

An equilibrium point is said to be asymptotically stable if for some initial value close to the equilibrium point, the solution will not only eventually stay close, but also converge to the equilibrium point.

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

Misc

For first order Differential Equations, the motion of a point in a Phase Portrait won’t cross a Equilibrium Point