Abstract:
We consider questions of non-uniqueness of cycles in phase portraits of systems of ordinary differential equations of biochemical kinetics with block-linear right-hand sides considered as models of simplest molecular repressilators functioning, and that of other circular gene networks. For these models of different dimensions, conditions of existence of cycles were elaborated previously and stability of these cycles was studied.
Now we describe a 3-dimensional dynamical system of this type and three piecewise linear cycles in its phase portrait, as well as their invariant neighborhoods, which are homeomorphic to a torus. This makes possible to localize these cycles and to determine
their mutual arrangement.
The smallest of these three cycles is an elementary example of a hidden attractor of a nonlinear dynamical system. Two remaining cycles give examples of non-local oscillations in the phase portrait.
Numerical experiments with this dynamical system illustrate the results. In the previous publications, non-uniqueness of cycles was detected in higher-dimensional cases only, starting from dim = 5.
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