Periodic Lyapunov differential equation for noise evaluation in oscillatory genetic networks

The Lyapunov equation, which describes stochastic variations of systems as a covariance equation, plays a central role in evaluation and prediction of behavior of genetic regulatory networks fluctuated by molecular noises. When the genetic network is an autonomously oscillatory system, the Lyapunov equation becomes a periodic differential equation that has generally unbounded solutions. We discuss orbital stability and periodicity properties of the solutions to the periodic Lyapunov differential equation by using Floquet-Lyapunov theory, and propose two global measures for evaluating stochastic fluctuations of the whole periodic trajectory. We also provide an evaluation formula for the period fluctuation.