Abstract
The necessary and sufficient conditions are obtained for the existence of a cycle in an odd-dimensional nonlinear dynamical system that simulates the functioning of the simplest circular gene network. An invariant neighborhood of this cycle is described which is homeomorphic to a torus.
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A. A. Akin’shin, V. P. Golubyatnikov, and I. V. Golubyatnikov, “On Some Multidimensional Models of Gene Networks Functioning,” Sibir. Zh. Industr. Mat. 16 (1), 3–9 (2013) [J. Appl. Indust. Math. 7 (3), 296–301 (2013)].
N. B. Ayupova and V. P. Golubyatnikov, “On theUniqueness of a Cycle in an Asymmetric Three-Dimensional Model of a Molecular Repressilator,” Sibir. Zh. Industr. Mat. 17 (1), 3–7 (2014) [J. Appl. Indust. Math. 8 (2), 153–157 (2014)].
V. P. Golubyatnikov and I. V. Golubyatnikov, “Geometry and Topology of the Phase Portraits of Glass–Pasternak Dynamical Systems of Small Dimension,” Mat. Struktury i Model. 31 (3), 40–47 (2014).
V. P. Golubyatnikov and I. V. Golubyatnikov, “On Multidimensional Models of Gene Network Functioning,” Sibir. Zh. Industr. Mat. 13 (3), 13–18 (2010) [J. Appl. Indust. Math. 5 (3), 343–347 (2011)].
V. P. Golubyatnikov, I. V. Golubyatnikov, and V. A. Likhoshvai, “On the Existence and Stability of Cycles in Five-Dimensional Models of Gene Networks,” Sibir. Zh. Vychisl. Mat. 13 (4), 403–411 (2010) [Numer. Anal. Appl. 3 (4), 329–335 (2010)].
M. V. Kazantsev, “On Some Properties of Domain Graphs of Dynamical Systems,” Sibir. Zh. Industr. Mat. 18 (4), 42–49 (2015).
A. Yu. Kolesov, N. Kh. Rozov, and V. A. Sadovnichii, “Periodic Solutions of a TravelingWave Type in Circular GeneNetworks,” Izv. Ross. Akad. Nauk. Ser. Mat. 80 (3), 67–94 (2016) [Izv. Math. 80 (3), 523–548 (2016)].
L. Glass and J. S. Pasternack, “Stable Oscillations inMathematicalModels of Biological Control Systems,” J. Math. Biol. 6, 207–223 (1978).
V. A. Likhoshvai, V. P. Golubyatnikov, G. V. Demidenko, A. A. Evdokimov, and S. I. Fadeev, “Theory of Gene Networks,” in Systemic Computer Biology (Izd. Sibir. Otdel. Ross. Akad. Nauk, Novosibirsk, 2008), pp. 395–480.
G. Yu. Riznichenko, Lectures on Mathematical Models in Biology (Regul. i Khaotich. Dinam., Izhevsk, 2011) [in Russian].
V. P. Golubyatnikov, V. V. Ivanov, and L. S. Minushkina, “OnExistence of aCycle in anNonsymmetricModel of a Circular Gene Network,” Sibir. Zh. Chist. Priklad. Mat. 18 (3), 26–32 (2018).
A. Tarski, “A Lattice Theoretical Fixed Point Theorem and Its Applications,” Pacific J. Math. 5, 285–309 (1955).
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Original Russian Text © V.P. Golubyatnikov, V.V. Ivanov, 2018, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2018, Vol. XXI, No. 4, pp. 28–38.
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Golubyatnikov, V.P., Ivanov, V.V. Cycles in the Odd-Dimensional Models of Circular Gene Networks. J. Appl. Ind. Math. 12, 648–657 (2018). https://doi.org/10.1134/S1990478918040051
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DOI: https://doi.org/10.1134/S1990478918040051