We show how to characterize a strange attractor by a set of integers. These are extracted from the chaotic time-series data by first reconstructing the low-period orbits and then determining the template, or knot holder, which supports all periodic orbits embedded in the strange attractor, and the strange attractor itself. The template is identified by a set of integers which therefore characterize the strange attractor. This identification is explicitly demonstrated for the Pirogon using a relatively small data set (5000 points).

We show that a qualitative theory of bifurcations can provide an understanding of the spontaneous symmetry breaking and the appearance of traveling waves leading to spatiotemporal complexity in the transverse patterns of the intensity of a ${\mathrm{CO}}_2$ laser.