Heteroclinic Interactions of Resonant Saddle

The figure llustrates the phase-space topology for a periodically-forced laser sampled at the forcing period. The outer ring illustrates a period-two subharmonic resonance that appears through a saddle-node bifurcation. The inner ring shows a period-one subharmonic resonance that has undergone a period-doubling bifurcation; the resulting saddle is a flip-saddle. The figure illustrates a forward-heteroclinic intersection (circled) of the period-two unstable manifold with the period-one flip-saddle's stable manifold. For higher forcing frequency a reverse-heteroclinic connection is also formed. The result is a connection between the period-one and period-two basins of attraction that leads to highly-complex transients and eventually chaos. This work appeared in T.W. Carr, L. Billings, I.B. Schwartz, and I. Triandaf, "Bi-instability and the global role of unstable resonant orbits in a driven laser," Physica D, 147:59-82, 2000.