Lyapunov functions for networks of asymmetrically iISS systems and circadian oscillations

This paper proposes new tools for stability analysis of networks of scalar dissipative systems. One of the aims of the development is to determine ranges of parameters which are necessary for maintaining robustly persistent circadian oscillations in the cells of Drosophila. This paper takes a Lyapunov-type analytical approach instead of numerical approaches which have been dominant in system-theoretic studies of circadian oscillations. Since each variable describing a substance in a biological network naturally involves nonlinear autonomous and interactive dynamics, the dissipativity is usually asymmetric with respect to a balance point. This paper shows how to make use of integral input-to-state stability (iISS) for a class of systems covering such biological networks. General tools to verify stability and explicitly construct Lyapunov functions are presented.