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Non-Uniqueness of Cycles in Piecewise-Linear Models of Circular Gene Networks


We find conditions for existence of two cycles for a five-dimensional piecewise-linear dynamical system that models functioning of a circular gene network. Conditions for existence of a cycle were obtained by the authors earlier. The phase portrait of a system is divided into subdomains (or blocks). With the use of such a discretization, we construct a combinatorial scheme for passages of trajectories between blocks. For the second cycle, we show that such a scheme depends on the parameters of a system.

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The authors are sincerely grateful to V. V. Ivanov for useful discussions and to the anonymous referee for her/his critical remarks.


The work was partially supported by the Russian Foundation for Basic Research (project 18-01-00057) and the Program of Fundamental Scientific Research of the SB RAS no. I.1.5 (project 0314-2018-0011).

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Correspondence to V. P. Golubyatnikov or V. S. Gradov.

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Golubyatnikov, V.P., Gradov, V.S. Non-Uniqueness of Cycles in Piecewise-Linear Models of Circular Gene Networks. Sib. Adv. Math. 31, 1–12 (2021).

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  • nonlinear dynamical systems
  • phase portraits
  • invariant domains
  • valency of a block
  • cycles