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On cycles in 6D circular gene networks models
We study a six-dimensional dynamical system with smooth functions simulating a circular gene network with positive and negative feedback and nonlinear degradation. An invariant domain for the trajectories of the dynamical system is constructed. The domain contains a unique equilibrium point. Using methods of qualitative analysis, we have found the sufficient condition of existence of a cycle in the invariant subdomain. Moreover, the trajectories of the dynamical systems lie on a two-dimensional invariant surface bounded by a cycle.