Figure 2
Eigenvalues 
of the Jacobian matrix 
correspond to intersections of the two function 
and 
. This Figure illustrates how a stable fixed point is destabilized by a time delay Ï„. (a) For 
, 
is a constant function, and the Jacobian matrix 
has two negative real eigenvalues 
and 
. (b) For 
, the function 
is a strictly increasing function that approaches 0 exponentially. Thus, increasing Ï„, the two real eigenvalues coalesce to a pair of complex conjugate eigenvalues, whose real parts eventually become positive. (c) A further increase in Ï„ leads to a new osculation point 
of 
and 
at a value 
. This 
is positive and real, and hence corresponds to an unstable fixed point. (d) For 
, the two eigenvalues approach the values 
and 
.