Target waves in the complex ginzburg-landau equation

We introduce a spatially localized inhomogeneity into the two-dimensional complex 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 propagating periodic traveling wave. In the breathing case, however, the region in the vicinity of the target center experiences a periodic temporal modulation at a frequency, in addition to that of the wave frequency of the faraway outward waves. Thus at a fixed point near the target, the breathing case yields a quasiperiodic time variation of the field. We investigate the transition between stationary and breathing targets, and note the existence of hysteresis. We also discuss the competition between the two types of target waves and spiral waves.