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Finding equilibrium points
. 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 is an equilibrium point for the following autonomous differential equations:

In other words:

In order to determine the stability of this equilibrium point we first find a linear approximation of the system near this point:
and then check the values of: and to determine stability. Note that or
are special cases and we cannot use the table to determine the stability.

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*Eran Borenstein *

2007-05-04