Periodic trajectories of nonlinear circular gene network models

The article is devoted to qualitative analysis of two dynamical systems simulating circular gene network functioning. The equations of three-dimensional dynamical system contain monotonically decreasing smooth functions describing negative feedback. A six-dimensional dynamical system consists of three equations with monotonically decreasing smooth functions and three equations with monotonically increasing smooth functions characterizing negative and positive feedback. In both models the process of degradation is described by nonlinear smooth functions. In order to localize cycles for both systems invariants domains are constructed. It is shown that each of two systems has a unique stationary point in the invariant domain, and conditions under which this point is hyperbolic are found. The main result of the paper is the proof of existence of a cycle in the invariant subdomain from which the trajectories can not pass to other subdomains obtained by discretization of the phase portrait. The cycles of three- and six-dimensional systems bound two-dimensional invariant surfaces, on which the trajectories of these dynamical systems lie.