The organization of multiple coexisting basins of attraction in two-dimensional driven dynamical systems is studied. This study is carried out, in particular, for the laser with modulated parameter and the Hénon map. Basin organization is governed primarily by the ordering of heteroclinic and homoclinic connections of regular saddles. This organization is complex yet systematic, with accumulation structures governed by the order of saddle connections. We study the evolution of a basin in the presence of multiple coexisting basins, from (before) birth to (beyond) death. Many of the processes that occur are canonical. These include the reorganization of existing basins in preparation for the creation of a new basin by saddle-node bifurcation, the increased intertwining, in Poincaré section, of the components of a basin, and the increased wrinkling of each component as the basin evolves from birth to death. These also include the processes that mark disintegration and the ultimate demise of the basin, beginning at homoclinic tangency and terminating in its disappearance or enlargement in a crisis.

• Received 18 April 1988

DOI:https://doi.org/10.1103/PhysRevA.39.2609