Phase portraits of a block-linear dynamical system in a model for a circular gene network

The article is devoted to a dynamical system with discontinuous functions on the right-hand sides of its equations that simulates functioning of a circular gene network with positive and negative feedback. In the first part of this paper we describe construction of an invariant domain and its splitting into subdomains containing blocks of different valence. We consider questions of existence, uniqueness and stability for the periodic trajectory and properties of the Poincaré map in the subdomain with blocks of minimal valence. In the second part, we study behavior and geometric features of trajectories contained in the non-invariant subdomain with blocks of maximal valence. We also describe the construction of a piecewise-linear invariant surface for a system symmetric up to cyclic permutations of variables. It is proved that this subdomain does not contain cycles of the dynamical system.