Delay-induced inward and outward spiral waves in oscillatory medium.

Inward and outward spiral waves as well as inward target waves are induced by local delay feedback in a reaction-diffusion system exhibiting a Turing hexagon pattern spontaneously. The system gives rise to large-amplitude sinusoidal oscillations when the inward spiral waves are observed. Compared with the inward spirals, the outward spiral waves usually possess longer wavelength and exhibit larger amplitude relaxation oscillations. Varying the feedback parameters continuously, the inward spiral waves are transformed into outward spirals through a turbulent state. The dispersion analysis about the delayed system reveals that there is an unstable band on the dispersion curve, where the turbulent state arises. The dispersion curve is divided into two parts by the unstable band. The inward spiral waves exist in the left part with negative group velocity, i.e., domega/dk<0, while the outward spiral waves are stable in the right part with domega/dk>0 (omega is frequency and k wavenumber).