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1.2.4.1 Stability and local analysis

Finding equilibrium points $ \dot{v}_1 = \dot{v}_2 = 0$. Sometimes we may be able to find equilibrium points analytically. In most cases however this is difficult if not impossible. However, we can find these by identifying the intersections of the nullclines. Moreover, in a tiny neighborhood of these intersections (equilibria points) the differential equation can be well approximated by linear equation. This is helpful because in this neighborhood we can therefore apply the analysis we studied for linear equations to determine the stability of these equilibria. Suppose that $ (e1,e2)$ is an equilibrium point for the following autonomous differential equations:
$\displaystyle \dot v_1 = F(v_1,v_2)$      
$\displaystyle \dot v_2 = G(v_1,v_2)$      

In other words:
$\displaystyle F(e_1,e_2) = 0$      
$\displaystyle G(e_1,e_2) = 0$      

In order to determine the stability of this equilibrium point $ (e1,e2)$ we first find a linear approximation of the system near this point:

\begin{displaymath}
\begin{array}{cc}
a = \frac{\partial F}{\partial v_1} \vert...
...= \frac{\partial G}{\partial v_2} \vert _{(e1,e2)}
\end{array}\end{displaymath}

and then check the values of: $ a+d$ and $ ad-bc$ to determine stability. Note that $ a+d = 0$ or $ ad - bc = 0$ are special cases and we cannot use the table to determine the stability.


next up previous contents
Next: 1.2.4.2 Two neurons example Up: 1.2.4 Pairs of non-linear Previous: 1.2.4 Pairs of non-linear
Eran Borenstein
2007-05-04