| In three dimensional dissipative dynamical systems with one stretching and one squeezing direction, chaos is often generated by stretching and folding processes. In a well-defined limit of high dissipation, the fold singularity A2 (logistic map) provides the backbone describing formation of a Smale horseshoe. In four dimensions with two stretching and one squeezing directions, and in the same well-defined limit, the cusp singularity A3 (cusp map) provides the backbone for processes creating chaos. Just as features of the logistic map, such as period doubling cascades and systematically sized windows, enable experimentalists to recognize the presence of three dimensional chaotic flows, the typical features of the cusp map will enable experimentalists to recognize the presence of higher dimensional flows with more than one unstable direction. In this work the systematic organization of "windows" associated with the cusp map is described and analyzed. |
