5. Frequency Conversion

Theoretical and experimental works on arrays of coupled oscillators have shown that symmetry alone can lead to periodic patterns with “multiple frequencies” of oscillation [15, 63, 144, 147, 387]. In Ref. [15], for instance, it is shown that a ring of N oscillators possessing \(\mathbf{D}_N\)-symmetry (symmetry of an N-gon) can induce, under certain conditions, an external oscillator to oscillate at N times the collective frequency of the ring. The actual conditions require that the ring oscillates in a traveling wave pattern and that the cross-coupling be directed from the ring to the external oscillator, i.e., a master-slave system. An alternative approach, which is pursued in this chapter, is to use the existing theoretical works in [144, 146, 147, 149], which show (within the context of a coupled cell system) that \(\mathbf{Z}_N\) symmetry-breaking Hopf bifurcations can also lead to multifrequency patterns. In particular, the multi-frequency patterns discussed in this chapter are motivated by an application: a “channelizer”. This is a broad spectrum analyzer that can instantaneously lock onto Radio Frequency (RF) signals of multiple frequencies, see Fig. 5.1.