Geodesic
On existence of a cycle in one asymmetric model of a molecular repressilator
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 2, pp. 121-129 
N. B. Ayupova; V. P. Golubyatnikov; M. V. Kazantsev. On existence of a cycle in one asymmetric model of a molecular repressilator. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 20 (2017) no. 2, pp. 121-129. http://geodesic.mathdoc.fr/item/SJVM_2017_20_2_a0/
@article{SJVM_2017_20_2_a0,
     author = {N. B. Ayupova and V. P. Golubyatnikov and M. V. Kazantsev},
     title = {On existence of a~cycle in one asymmetric model of a~molecular repressilator},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {121--129},
     year = {2017},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2017_20_2_a0/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

We consider a nonlinear $6$-dimensional dynamic system which is a model of functioning of one simple molecular repressilator and find sufficient conditions of existence of a cycle $\mathcal C$ in the phase portrait of this system. An invariant neighborhood of $\mathcal C$ which retracts to $\mathcal C$ has been constructed.

[1] Elowitz M. B., Leibler S., “A synthetic oscillatory network of transcriptional regulators”, Nature, 403 (2000), 335–338 | DOI

[2] Glyzin S. D., Kolesov A. Yu., Rozov N. Kh., “Yavlenie bufernosti v koltsevykh gennykh setyakh”, Teoreticheskaya i matematicheskaya fizika, 187:3 (2016), 560–579 | DOI | MR | Zbl

[3] Kolesov A. Yu., Rozov N. Kh., Sadovnichii V. A., “Periodicheskie resheniya tipa beguschikh voln v koltsevykh gennykh setyakh”, Izvestiya RAN, seriya matematicheskaya, 80:3 (2016), 67–94 | DOI | MR | Zbl

[4] Likhoshvai V. A., Matushkin Yu. G., Fadeev S. I., “Zadachi funktsionirovaniya gennykh setei”, Sib. zhurn. industrialnoi matematiki, 6:2 (2003), 64–80 | MR | Zbl

[5] Demidenko G. V., Kolchanov N. A., Likhoshvai V. A., Matushkin Yu. G., Fadeev S. I., “Matematicheskoe modelirovanie regulyatornykh konturov gennykh setei”, Zhurn. vychisl. matem. i mat. fiziki, 44:12 (2004), 2276–2295 | MR | Zbl

[6] Likhoshvai V. A., Fadeev S. I., Demidenko G. V., Matushkin Yu. G., “Modelirovanie uravneniem s zapazdyvayuschim argumentom mnogostadiinogo sinteza bez vetvleniya”, Sib. zhurn. industrialnoi matematiki, 7:1 (2004), 73–94 | MR | Zbl

[7] Golubyatnikov V. P., Gaidov Yu. A., Kleshchev A. G., Volokitin E. P., “Modeling of asymmetric gene networks functioning with different types of regulation”, Biophysics, 51, suppl. 1 (2006), 61–65 | DOI

[8] Golubyatnikov V. P., Kleschev A. G., Klescheva K. A., Kudryavtseva A. V., “Issledovaniya fazovykh portretov trekhmernykh modelei gennykh setei”, Sib. zhurn. industrialnoi matematiki, 9:1 (2006), 75–84 | MR | Zbl

[9] Gedeon T., Cyclic feedback systems, Memoirs of American Math. Society, 134, no. 637, 1998, 73 pp. | DOI | MR

[10] Hirsch M., “Systems of differential equations which are competitive or cooperative. I: Limit sets”, SIAM J. Math. Anal., 13 (1982), 167–179 | DOI | MR | Zbl

[11] Hastings S., Tyson J. J., Webster D., “Existence of periodic solutions for negative feedbacks cellular control systems”, J. Diff. Equations, 25 (1977), 39–64 | DOI | MR | Zbl

[12] Golubyatnikov V. P., Golubyatnikov I. V., “On periodic trajectories in odd-dimensional gene networks models”, Russian J. of numerical analysis and mathematical modeling, 28:4 (2011), 397–412 | MR

[13] Ayupova N. B., Golubyatnikov V. P., “O dvukh klassakh nelineinykh dinamicheskikh sistem. Chetyrekhmernyi sluchai”, Sib. mat. zhurn., 56:2 (2015), 282–289 | MR | Zbl

[14] Glass L., Pasternack J. S., “Stable oscillations in mathematical models of biological control systems”, J. of Mathematical Biology, 6 (1978), 207–223 | DOI | MR | Zbl

[15] Ayupova N. B., Golubyatnikov V. P., “O edinstvennosti tsikla v trekhmernoi modeli molekulyarnogo repressilyatora”, Sib. zhurn. industrialnoi matematiki, 17:1 (2014), 3–7 | MR | Zbl

[16] Golubyatnikov V. P., Golubyatnikov I. V., Likhoshvai V. A., “O suschestvovanii i ustoichivosti tsiklov v pyatimernykh modelyakh gennykh setei”, Sib. zhurn. vychisl. matematiki (Novosibirsk), 13:4 (2010), 403–411 | Zbl

[17] Golubyatnikov V. P., Golubyatnikov I. V., “O periodicheskikh traektoriyakh nelineinykh dinamicheskikh sistem spetsialnogo vida”, Vestnik NGU. Seriya: Matematika, mekhanika, informatika, 10:3 (2010), 3–16 | Zbl

[18] Akinshin A. A., Golubyatnikov V. P., “Tsikly v simmetrichnykh dinamicheskikh sistemakh”, Vestnik NGU, 12:2 (2012), 3–12 | Zbl

[19] Kazantsev M. V., “O nekotorykh svoistvakh grafov domenov dinamicheskikh sistem”, Sib. zhurn. industrialnoi matematiki, 18:4 (2015), 42–48 | DOI | MR | Zbl

[20] Petrovskii I. G., Lektsii po teorii obyknovennykh differentsialnykh uravneniyakh, Izd-e 7-e, MGU, M., 1984 | MR

[21] Gaidov Yu. A., Golubyatnikov V. P., “On cycles and other geometric phenomena in phase portraits of some nonlinear dynamical systems”, Geometry and Applications, Springer Proc. in Mathematics Statistics, 72, Springer, NY, 2014, 225–233 | DOI | MR | Zbl