Target waves in the complex Ginzburg-Landau equation

Abstract

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.

Target patterns in two-dimensional heterogeneous oscillatory reaction–diffusion systems

“…For another modified version of the CGLE, properties of heterogeneous pacemakers have been studied by Hendrey and co-authors [7]. They assume that the heterogeneity not only changes the frequency, but also the amplitude of oscillations.…”

“…Most basic features of such patterns already were described by the beginning of the 1980s [3][4][5][6], but since then target patterns have received less attention than, e.g., rotating spiral waves. Recently, renewed interest in target patterns and pacemakers has yielded considerable progress in the understanding of these patterns [7][8][9][10][11][12][13].…”

Dynamics of defects in the vector complex Ginzburg–Landau equation

“…In that case, the phase of the nonvanishing field component presents a clearer target pattern. Phase target patterns in the scalar CGL equation are also found under suitable boundary conditions [28,29] or inhomogeneities [30]. The amplitudes and phases of the x and y linearly polarized components of the field around a scalar defect, also shown in Figs.…”

Spiral Waves in Nonlocal Equations

“…These are commonly observed in systems which also support spiral waves [16] and have been seen in simulations of large networks of spiking neurons [19]. In order for these waves to continue to be emitted from the center of the domain, some inhomogeneity must be present there.…”