Many living systems are regulated by a circadian clock that oscillates with a constant period of 24 hours. The circadian clock of the cyanobacteria S. elongates consists of a rather simple biochemical network consisting of just three proteins: KaiA, KaiB, and KaiC. We have previously used small catalytic reaction networks as models for producing oscillatory behavior, and present here a ternary network of coupled oscillators in order to mimic the connectivity and network topology of these Kai proteins. We show that such a synthetic network functions effectively as an internal clock, capable of oscillating with constant frequency, independent of the input intake rate, while exhibiting robust behavior with respect to fluctuations in initial concentrations. Furthermore, we compare our proposed circadian network with several alternate network topologies, and find our proposed network to be optimal and robust under a wider set of conditions. We suggest that our simulation and theoretical analysis helps further reveal the underlying principles of biological clocks, and may provide clues towards their emergence in early molecular evolution.