EQUILIBRIUM POINTS

Let the vector be the state of the dynamical system. The function tells us how the system moves, e.g.

In special circumstances, however, the system does not move. The system can be stuck (we shall say fixed) in a special state. We call these states fixed (equilibrium) points of the dynamical system.

- Worked Example 1

Consider the system

This may be rewritten as where

In order to find equilibrium points we solve the equations

from which one can find

The equilibrium points can be stable, unstable or semistable. Suppose a first order dynamical system has an equilibrium value This equilibrium value is said to be stable or attracting if there is a number , unique to each system, such that, when

then

An equilibrium value is unstable or repelling if there is a number such that, when

then

for some, but not necessarily all, values of Note that is a measure of the distance between the two values and The definition states that an equilibrium value is stable if whenever is sufficiently close to then tends to

The equilibrium point is called marginally stable or neutral provided the following: for all starting values near the system stays near but does not converge to .

Consider a dynamical system (called a general affine system)

Theorem:

The equilibrium value for this dynamical system is stable, if for any value of Also, if then the equilibrium value is unstable and goes to infinity for any

The equilibrium value is called semistable if for some values of tends to while for others goes away from