Uniqueness and stability of a cycle in three-dimensional block-linear circular gene network models

We study dynamics of three-dimensional autonomous systems which model circular gene network functioning regulated by negative feedbacks realized by step-functions. We prove that such a system has at most one cycle, and if this cycle does exist, then it is stable in Lyapunov's sense and it attracts all trajectories of the system except for two of them, which pass through the singular point of the system. The proofs of the main results reduce to studies of existence, uniqueness and geometry structure of the second fixed point of 2-dimensional monotonic mapping with monotonic derivatives.