# Dragon Notes

UNDER CONSTRUCTION
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# Nonlinear Dynamics & Chaos:Criticality

Stable Node

A node into which trajectories decay
as $$t\rightarrow\infty$$

\ds \begin{align} \dot{x} &= ax \\ \dot{y} &= -y;\ \ a<-1 \end{align}
Star Node

A stable node with equal decay rates in
both directions $$(\dot{x}=\dot{y})$$

\ds \begin{align} \dot{x} &= ax \\ \dot{y} &= -y;\ \ a=-1 \end{align}
Degenerate Node

Occurs in linear systems with one eigendirection
$$\ds A=\mtxx{\lambda}{b}{0}{\lambda},\ b\neq 0$$
A node with $$\pm$$eigenvectors along which trajectories exponentially grow and decay
\ds \begin{align} \dot{x} &= ax \\ \dot{y} &= -y;\ \ a>0 \end{align}
\ds \begin{align} \dot{x} &= v \\ \dot{v} &= -\omega^2 x;\ \ \omega = 2 \end{align}