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Linearising a System

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.

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