An equilibrium point to a system

i said to be *isolated* if there is a
neighborhood to the critical point that does not any other critical
points. There are four different types of isolated critical points that
usually occur. They are *center, node, saddle point *and *spiral*.

An equilibrium point can be*stable*, *asymptotical stabl*e
or *unstable*. A point is stable if the orbit of the system is
inside a bounded neighborhood to the point for all times *t*
after some *t*_{0}. A point is
aymptotical stable if it is stable and the orbit approaches the
critical point as
. If a critical point is not stable then it is unstable. In the figure
above we see that a center is stable but not asymptotically stable,
that a saddle point is unstable, that a node is either asymptotically
stable (sink) or unstable (source) and that a spiral either is
asymptotically stable or unstable.

In certain nonlinear systems we might also have
"mixtures" of the above types for *higher order critical points*.
See the example above.

An equilibrium point can be

Previous part Back to chapter 9: Dynamical systems, chaos, stability and bifurcations. Next part