The interaction of two chaotic rotating waves of the type recently reported by Matı́as et al. [Europhys. Lett. 37 (1997) 379] is studied experimentally with arrays of non-linear electronic circuits arranged in ring geometries. Unidirectional coupling is assumed for the cell-to-cell coupling within the same ring, but between rings, cells are coupled diffusively. Depending on the relative sense of driving, competition between a rotating chaotic wave and a global synchronized state has been observed. The results are rationalized by means of a linear stability analysis around the uniform synchronized behavior, where the circulant symmetry of the system allows to express the problem as a superposition of a series of Fourier modes.