Classification of fixed points from a system of equations

Classification if both Eigenvalues are real and positive?

Unstable Node.

Could be asked to plot this.

Classification if both eigenvalues are real and negative?

Stable Node.

Could be asked to plot this.

Classification if 0<\lambda_1=\lambda_2 and $A=cI$

It is a unstable star.

Could be asked to plot this

Classification if 0<\lambda_1=\lambda_2 and $A\neq cI$

It is an improper node

Unstable

Classification if \lambda_1=\lambda_2<0 and $A=cI$

Stable star.

Could be asked to plot this.

Classification if \lambda_1=\lambda_2<0 and $A\neq cI$

Stable Improper Node

Classification if \lambda_1>0,\lambda_2<0

Unstable saddle point.

Could be asked to draw this

Classification if $\lambda=\pm i\mu$

Stable centre

Classification if \lambda=k\pm i\mu, k>0

Unstable spiral

Classification if \lambda=k\pm i\mu , k<0

Stable spiral

How do you classify a fixed point of a nonlinear system of ODEs using the Jacobian?

For $\dot{x}=f(x,y),\ \dot{y}=g(x,y)$, first find fixed points by solving $f(x_0,y_0)=g(x_0,y_0)=0$. Then form the Jacobian $J=\begin{pmatrix}f_x&amp;f_y\\g_x&amp;g_y\end{pmatrix}$ and evaluate it at $(x_0,y_0)$.

Classify the fixed point using the eigenvalues of $J(x_0,y_0)$

NOTE: purely imaginary = centre in the linearised system, but for nonlinear systems it may become a centre or a spiral.

How do you linearise a nonlinear system near a stationary point $(x_0, y_0)$?

Taylor expand $f$ and $g$ about $(x_0, y_0)$, dropping higher order terms.

Set $X = x - x_0$, $Y = y - y_0$ to get the linearised system: $\frac{d}{dt}\begin{bmatrix} X \\ Y \end{bmatrix} = J(x_0, y_0)\begin{bmatrix} X \\ Y \end{bmatrix}$ where $J(x_0, y_0)$ is the Jacobian evaluated at the stationary point.