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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A. A. Akinshina and V. P. Golubyatnikov, “Geometric characteristics of cycles in some symmetric dynamical systems,” Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform. 12:2, 3 (2012) [in Russian].
N. B. Ayupova and V. P. Golubyatnikov, “On the uniqueness of a cycle in an asymmetric three-dimensional model of a molecular repressilator,” Sib. Zh. Ind. Mat. 17, 3 (2014) [J. Appl. Ind. Math. 8, 153 (2014)].
N. B. Ayupova and V. P. Golubyatnikov, “On two classes of nonlinear dynamical systems: the four-dimensional case,” Sib. Matem. Zh. 56, 282 (2015) [Siberian Math. J. 56, 231 (2015)].
H. T. Banks and J. M. Mahaffy, “Stability of cyclic gene models for systems involving repression,” J. Theor. Biology 74, 323 (1978).
Yu. A. Gaĭdov and V. P. Golubyatnikov, “On some nonlinear dynamical systems modeling asymmetric gene networks,” Vestn. Novosib. Gos. Univ., Ser. Mat. Mekh. Inform. 7:2, 19 (2007) [in Russian].
Yu. A. Gaĭdov, V. P. Golubyatnikov, A. G. Kleshchev, and E. P. Volokitin, “Modeling of asymmetric gene networks functioning with different types of regulation,” Biophysics 51, 61 (2006).
L. Glass and J. S. Pasternack, “Stable oscillations in mathematical models of biological control systems,” J. Math. Biology 6, 207 (1978).
V. P. Golubyatnikov, I. V. Golubyatnikov, and V. A. Likhoshvaĭ, “On the existence and stability of cycles in \(5\)-dimensional models of gene networks,” Sib. Zh. Vychisl. Mat. 13, 403 (2010) [Numer. Analysis Appl. 3, 329 (2010)].
V. P. Golubyatnikov and V. V. Ivanov, “Cycles in the odd-dimensional models of circular gene networks,” Sib. Zh. Ind. Mat. 21, no. 4, 28 (2018) [J. Appl. Ind. Math. 12, 648 (2018)].
V. P. Golubyatnikov and V. V. Ivanov, “Uniqueness and stability of a cycle in three-dimensional block-linear circular gene network models,” Sib. Zh. Chist. Prikl. Mat. 18:4, 19 (2018) [in Russian].
V. P. Golubyatnikov and M. V. Kazantsev, “Piecewise linear dynamical system modeling gene network with variable feedback,” Sib. Zh. Chist. Prikl. Mat. 16, no. 4, 28 (2016) [J. Math. Sci., New York 230, 46 (2018)].
S. Hastings, J. Tyson, and D. Webster, “Existence of periodic solutions for negative feedbacks cellular control systems,” J. Differ. Equations 25, 39 (1977).
M. V. Kazantsev, “On some properties of the domain graphs of dynamical systems,” Sib. Zh. Ind. Mat. 18:4, 42 (2015) [in Russian].
A. Yu. Kolesov, N. Kh. Rozov, and V. A. Sadovnichiĭ, “Periodic solutions of travelling-wave type in circular gene networks,” Izv. Ross. Akad. Nauk, Ser. Mat. 80, no. 3, 67 (2016) [Izv. Math. 80, 523 (2016)].
A. G. Kurosh, A Course in Higher Algebra (Nauka, Moscow, 1975) [in Russian].
V. A. Likhoshvaĭ, S. I. Fadeev, V. V. Kogaĭ, and T. M. Khlebodarova, “On the chaos in gene networks,” J. Bioinform. Comput. Biology 11:1, 1340009 (2013).
G. Yu. Riznichenko, Lectures on Mathematical Models in Biology, (Scientific Research Center “Regular and Chaotic Dynamics”, Moscow–Izhevsk, 2002) [in Russian].
S. Tabachnikov, Geometry and Billiards (Amer. Math. Soc., Providence, RI, 2005).
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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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). https://doi.org/10.1134/S1055134421010016
- nonlinear dynamical systems
- phase portraits
- invariant domains
- valency of a block