Classification of Critical Points

Suppose we have a system of coupled differential equations
\[\dot{x}=F(x,y)\]

\[\dot{y}=G(x,y)\]

This system may be very difficult or impossible to solve, but we can find the nature of the solution about a point  
\[(x_0,y_0)\]
  by linearising the system about the point  
\[(x_0,y_0)\]

\[\dot{x}=\frac{\partial F}{\partial x}|_{(x_0, y_0)}(x-x_0) +\frac{\partial F}{\partial y}|_{(x_0, y_0)}(y-y_0) + higher \: order \: terms\]

\[\dot{y}=\frac{\partial G}{\partial x}|_{(x_0, y_0)}(x-x_0) +\frac{\partial G}{\partial y}|_{(x_0, y_0)}(y-y_0) + higher \: order \: terms\]

We can define  
\[x'=x-x_0, \: y'=y-y_0\]
  and the system becomes, in matrix form:
\[\begin{pmatrix}\dot{x}'\\ \dot{y}'\end{pmatrix} = \left( \begin{array}{cc} \frac{\partial F}{\partial x}|_{(x_0, y_0)} & \frac{\partial F}{\partial y}|_{(x_0, y_0)} \\ \frac{\partial G}{\partial x}|_{(x_0, y_0)} & \frac{\partial G}{\partial y}|_{(x_0, y_0)} \end{array} \right) \begin{pmatrix}x'\\ y'\end{pmatrix} \]

We are especially interested in the point(s)  
\[(x_0,y_0)\]
  that satisfy  
\[\dot{x}=\dot{y}=0\]
. These are called critical points.
The linearisation matrix  
\[\left( \begin{array}{cc} \frac{\partial F}{\partial x}|_{(x_0, y_0)} & \frac{\partial F}{\partial y}|_{(x_0, y_0)} \\ \frac{\partial G}{\partial x}|_{(x_0, y_0)} & \frac{\partial G}{\partial y}|_{(x_0, y_0)} \end{array} \right)\]
  has eigenavlues  
\[\lambda_1 , \: \lambda_2\]
  and corresponding eigenvectors  
\[\mathbf{v}_1, \: \mathbf{v}_2\]
.
If  
\[\lambda_1 \: \lambda_2 \lt 0\]
  then the point  
\[(x_0, y_0)\]
  is stable. Any initial point on the eigenline (a line at starting at the point  
\[(x_0, y_0)\]
  in the direction of an eigenvector) will tend to the point  
\[(x_0, y_0)\]
  along that eigenline. Any initial point not on an eigenline will tend to the eigenline will the eigenline with the corresponding most negative eigenvalue, then towards the point  
\[(x_0, y_0)\]
. This is a stable node.
If  
\[\lambda_1 \: \lambda_2 \gt 0\]
  then the point  
\[(x_0, y_0)\]
  is unstable. Any initial point on the eigenline will tend to move away from the point  
\[(x_0, y_0)\]
  along that eigenline. Any initial point not on an eigenline will tend to the eigenline will the eigenline with the corresponding most positive eigenvalue, then away from the point  
\[(x_0, y_0)\]
. This is an unstable node.
If  
\[\lambda_1 \gt 0, \: \lambda_2 \,t 0\]
  then the point  
\[(x_0, y_0)\]
  is unstable. Any initial point not on the eigenline corresponding to  
\[\lambda_2\]
  will tend to the point  
\[(x_0, y_0)\]
.Any other initial point on the eigenline corresponding to  
\[\lambda_1\]
  will tend to move away from the point  
\[(x_0, y_0)\]
  along that eigenline, or will tend to that eigenline away from the point  
\[(x_0, y_0)\]
.This is a saddle point.
If the eigenvalues are complex  
\[\lambda_1=a+bi, \: \lambda_2 =a-bi\]
  then if  
\[a \lt 0\]
  the point  
\[(x_0, y_0)\]
  is stable (a stable spiral) and unstable (an unstable spiral)if  
\[a \gt 0\]
. If  
\[a = 0\]
  then any point will move in a circle about the point  
\[(x_0, y_0)\]
. The point  
\[(x_0, y_0)\]
  is a centre.
If there is only one eigenvalue  
\[\lambda\]
  and only one eigenvector  
\[\mathbf{v}\]
  then the point  
\[(x_0,y_0)\]
  is a stable degenerate node if  
\[\lambda \lt 0\]
  and an unstable degenerate node if  
\[\lambda \gt 0\]
.

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