The first important characteristic of this system involves a Hopf bifurcation,
at which stable periodic solutions
emerge. A supercritical Hopf bifurcation occurs at 
,
as seen in Figure [8(c)].
With values of 
less than that at which a Hopf bifurcation occurs the
system exhibits stable behavior; all
trajectories settle to a fixed point in the system (Figure [9(a)]).
Figure 9: Er vs Time. (a) Stable fixed point. (b) Stable periodic solutions. (c) Chaotic regime.
At just above the Hopf
bifurcation, all solutions tend to the stable periodic solution (Figure [9(b)]).
Immediately following the Hopf bifurcation, the system rapidly goes
through a series of period doublings. These periodic solutions remain
stable for values of up to 
. Following this, the system becomes chaotic (Figure [9(c)]); essentially
the period of the periodic solutions
has become infinite.
As b is further increased, a stable periodic solution emerges, at ( 
).
This is indicated by the solid filled circles on
the upper branch in Figure [8(b)].
All
trajectories in this region of b tend toward these periodic
solutions.
With b further increased, to 
, chaotic behavior emerges
again. The stable periodic solution
becomes unstable. Now, once again, there are no stable solutions and the
system behaves chaotically. This chaotic
behavior continues until 
, where stable periodic
solutions again occur. This behavior continues
until 
, at which point all trajectories settle
down to a stable fixed point; essentially the
behavior of the laser becomes dominated by the injection term .
A graphical summary of the behavior of the system as the injection paramater is varied can be seen in Figure [10]. In effect the addition of a constant injection term has created several regions of chaotic behavior, as well as several regions of periodic behavior, in a system which would otherwise possess no such dynamics.
Figure 10: Schematic Diagram of behavior of system versus injection parameter b.
Acknowledgements
This research was supported by an REU Supplement to
NSF Grant DMS-9626306, and an ASSERT Grant from AFOSR. This project was
part of the Undergraduate Research Initiative of the University of Arizona
Math Center. The authors would also like to thank Aaron King for permitting
them to use his normal form software.