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Behavior of trajectories

The first important characteristic of this system involves a Hopf bifurcation, at which stable periodic solutions emerge. A supercritical Hopf bifurcation occurs at tex2html_wrap1665 , as seen in Figure [8(c)].

With values of tex2html_wrap1603 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 tex2html_wrap1603 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 tex2html_wrap1603 up to tex2html_wrap1684 . 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 ( tex2html_wrap1685 ). 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 tex2html_wrap1686 , 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 tex2html_wrap1687 , where stable periodic solutions again occur. This behavior continues until tex2html_wrap1688 , at which point all trajectories settle down to a stable fixed point; essentially the behavior of the laser becomes dominated by the injection term tex2html_wrap1603 .

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.

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.

next up previous
Next: About this document Up: Laser Rate Equations With Previous: Behavior of fixed points