The nature of attractor basins in multistable systems

Abstract

In systems that exhibit multistability, namely those that have more than one coexisting attractor, the basins of attraction evolve in specific ways with the creation of each new attractor. These multiple attractors can be created via different mechanisms. When an attractor is formed via a saddle-node bifurcation, the size of its basin increases as a power-law in the bifurcation parameter. In systems with weak dissipation, the basins of low-order periodic attractors increase linearly, while those of high-order periodic attractors decay exponentially as the dissipation is increased. These general features are illustrated for autonomous as well as driven mappings. In addition, the boundaries of the basins can also change from being smooth to fractal when a new attractor appears. Transitions in the basin boundary morphology are reflected in abrupt changes in the dependence of the uncertainty exponent on the bifurcation parameter.