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In this paper, we aim to develop a new methodology to model and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using multiple timescale networks (MTN). Fast reactions constitute a positive feedback-loop network (PFN), while slow reactions consist of a cyclic feedback-loop network (CFN), in MTN. Multiple timescales are exploited to simplify models according to singular perturbation theory. We show that a MTN has no stable equilibrium but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of MTNs with only one PFN, and then generalise the result to MTNs with multiple PFNs. Finally, we design a biologically plausible gene regulatory network by the cI and Lac genes, to demonstrate the theoretical results. Since there is less restriction on the network structure of a MTN, it can be expected to apply to a wide variety of areas on the modelling, analysing and designing of biological systems.

This paper treats a nonlinear dynamical system with both continuous-time and discrete-time variables as a differential-difference-algebraic equation (DDA) or a hybrid dynamical system, presents a fundamental analyzing method of such a DDA system for local sampling, asymptotical stability, singular perturbations and bifurcations, and further shows that there exist four types of generic codimension-one bifurcations at the equilibria in contrast to two types in continuous-time dynamical systems and three types in discrete-time dynamical systems. Finally the theoretical results are applied to digital control of power systems as an example. Numerical simulations demonstrate that our results are useful.