Next: Up: Fixed Points Previous: Eigenvalues

### =-1

Assume . Solving (9) for yields

 (10)

In other words, using (7), the fixed point changes stability when . It can be seen from 7 that this is in fact the point at which the fixed points come into existence. Figure 1 is a bifurcation diagram of this system, plotting the stable fixed points of the system versus . The emergence of fixed points and changes in their stability is discussed in terms of increasing the parameter .

To examine the stability of the fixed points which emerge here, consider the system (1) without delay. We know that

 (11)

Once the points are born, it can be seen from (7) that one fixed point will increase in magnitude as increases, and the other point will decrease. For the point that increases in magnitude, (11) shows that its eigenvalue will be of magnitude less than 1 thus this point will be stable. For the other point, will become increasingly large in magnitude, thus this point will remain unstable.

Next: Up: Fixed Points Previous: Eigenvalues
URA Program Website 2004-12-02