An Elementary Proof for the Existence of Limit Cycle in Three Gene Cyclic Regulatory Network

Oscillatory networks are important in several biological contexts. However, establishing simple conditions for their existence is challenging, especially in higher dimensional models. Here, we apply an analytical technique to provide an elementary proof for the existence of oscillation in a benchmark cyclic three-dimensional oscillator. This analytical technique is based on the Brouwer's fixed point theorem. We establish the presence of a positively invariant region. A subset of this region is torus-like with the property that the flow maps a cross-section of the torus into itself. This leads to the existence of a limit cycle solution. These results should help in tracing the path of the limit cycle in the phase space.