next up previous
Next: Up: Fixed Points Previous: Eigenvalues

$\lambda $=-1

Assume $\lambda=-1$. Solving (9) for $x$ yields


$\displaystyle x = -\frac{1}{2}$     (10)

In other words, using (7), the fixed point changes stability when $a= -\frac{1}{4}$. 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 $a$. The emergence of fixed points and changes in their stability is discussed in terms of increasing the parameter $a$.

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


$\displaystyle \lambda = -2x = -1 \pm \sqrt{1+4a}$     (11)

Once the points are born, it can be seen from (7) that one fixed point will increase in magnitude as $a$ 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, $\lambda $ will become increasingly large in magnitude, thus this point will remain unstable.


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