Feedback loops are found to be important network structures in
regulatory networks of biological signaling systems because they are
responsible for maintaining normal cellular activity. Recently,
the functions of feedback loops have received extensive attention.
The existing results in the literature mainly focus on verifying
that negative feedback loops are responsible for oscillations,
positive feedback loops for multistability, and coupled feedback
loops for the combined dynamics observed in their individual loops.
In this work, we develop a general framework for studying systematically
functions of feedback loops networks. We investigate the general
dynamics of all networks with one to three nodes and one to two feedback loops. Interestingly, our results are consistent with Thomas' conjectures although we assume each node in the network undergoes a decay, which corresponds to a negative loop in Thomas' setting. Besides studying how network structures influence dynamics at the linear level, we explore the
possibility of network structures having impact on the nonlinear
dynamical behavior by using Lyapunov-Schmidt reduction and singularity theory.
Mathematics Subject Classification: Primary: 37N25, 37G10; Secondary: 34C23, 34A34.