Target waves in the complex Ginzburg-Landau equation

We introduce a spatially localized inhomogeneity into the two-dimensional c omplex Ginzburg-Landau equation. We observe that this can produce two types of target wave patterns: stationary and breathing. In both cases, far from the target center, the field variables correspond to an outward propagatin g periodic traveling wave. In the breathing case, however, the region in th e vicinity of the target center experiences a periodic temporal modulation at a frequency, in addition to that of the wave frequency of the faraway ou tward waves. Thus at a fixed point near the target, the breathing case yiel ds a quasiperiodic time variation of the field. We investigate the transiti on between stationary and breathing targets, and note the existence of hyst eresis. We also discuss the competition between the two types of target wav es and spiral waves.