Phase portraits of two nonlinear models of circular gene networks
https://doi.org/10.25587/SVFU.2023.54.12.001
Abstract
For two dynamical systems of dimensions 4 and 5 which simulate circular gene networks with non-linear degradation of their components we find conditions for existence of periodic trajectories and construct invariant domains which contain all these trajectories. Interiors of both domains are homeomorphic to torus, and the boundary of each of them contains a unique equilibrium point of the corresponding dynamical system.
About the Authors
N. B. Ayupova
Sobolev Institute of Mathematics
Russian Federation
Russian Federation
Natalya B. Ayupova
4 Koptyug Avenue, Novosibirsk 630090, Russia
V. P. Golubaytnikov
Sobolev Institute of Mathematics; Novosibirsk Military Institute of Russian National Guard Forces
Russian Federation
Russian Federation
Vladimir P. Golubyatnikov
4 Koptyug Avenue, Novosibirsk 630090, Russia
4/2 Klyuch-Kamyshenskoe Plateau, Novosibirsk 630114, Russia
References
1. Glass L. and Pasternack J. S., “Stable oscillations in mathematical models of biological control systems,” J. Math. Biol., 6, 207–223 (1978).
2. Likhoshvai V. A., Kogai V. V., Fadeev S. I., and Khlebodarova T. M., “On the chaos in gene networks,” J. Bioinform. Comput. Biol., 11, article No. 1340009 (2013).
3. Gedeon T., Pernarowski M., and Wilander A., “Cyclic feedback systems with quorum sensing coupling,” Bull. Math. Biol., 78, No. 6, 1291–1317 (2016).
4. Glyzin S. D., Kolesov A. Yu., and Rozov N. Kh., “Existence and stability of the relaxation cycle in a mathematical repressilator model,” Math. Notes, 101, No. 1, 71–86 (2017).
5. Hastings S., Tyson J., and Webster D., “Existence of periodic solutions for negative feedback cellular control system,” J. Differ. Equ., 25, 39–64 (1977).
6. Ayupova N. B. and Golubyatnikov V. P., “On two classes of non-linear dynamical systems: the four-dimensional case,” Sib. Math. J., 56, No. 2, 231–236 (2015).
7. Golubyatnikov V. P. and Gradov V. S., “Non-uniqueness of cycles in piecewise-linear models of circular gene networks,” Sib. Adv. Math., 31, No. 1, 1–12 (2021).
8. Golubyatnikov V. P. and Minushkina L. S., “On uniqueness and stability of a cycle in one gene network,” Sib. Electron. Math. Rep., 1, No. 1, 464–473 (2021).
9. Ayupova N. B. and Golubyatnikov V. P., “On structure of phase portrait of one 5-dimensional circular gene network model,” J. Appl. Ind. Math., 15, No. 3, 376–383 (2021).
10. Golubyatnikov V. P., Golubyatnikov I. V., and Likhoshvai V. A., “On the existence and stability of cycles in five-dimensional models of gene networks,” Numer. Anal. Appl., 3, No. 4, 329–335 (2010).
11. Gantmacher F. R., The Theory of Matrices, vol. 1, Chelsea Publ. Co. (1984).
12. Khlebodarova T. M., Kogai V. V., Fadeev S. I., and Likhoshvai V. A., “Chaos and hyperchaos in simple gene network with negative feedback and time delays,” J. Bioinform. Comput. Biol., 15, No. 2, article No. 1650042 (2017).
13. Arnold V. I., Gusein-Zade S. M., and Varchenko A. N., Singularities of Differentiable Maps, Birkh¨auser, Boston (1985).
14. Gaidov Yu. V., “On the stability of periodic trajectories in some gene network models,” J. Appl. Ind. Math., 4, No. 1, 43–47 (2010).
15. Kazantsev M. V., “On some properties of the domain graphs of dynamical systems [in Russian],” Sib. Zh. Ind. Mat., 56, No. 4, 42–48 (2015).
16. Golubyatnikov V. P. and Kirillova N. E., “Phase portraits of two gene networks models [in Russian],” Mat. Zamet. SVFU, 28, No. 1, 3–11 (2021).
17. Ivanov V. V., “Attracting limit cycle of an odd-dimensional circular gene network model,” J. Appl. Ind. Math., 16, No. 3, 409–415 (2022).
18. Kirillova N. E., “On invariant surfaces in gene network models,” J. Appl. Ind. Math., 14, No. 4, 666–671 (2020).
19. Golubyatnikov V. P., Likhoshvai V. A., and Ratushny A. V., “Existence of closed trajectories in 3-D gene networks,” J. Three Dimensional Images 3D Forum, Japan, 18, No. 4, 96–101 (2004).
20. Petrovskii I. G., Lectures on the Theory of Ordinary Differential Equations [in Russian], Nauka, Moscow (1970).
21. Hartman Ph., Ordinary Differential Equations, 2nd ed., SIAM (1982).
22. Chumakov G. A. and Chumakova N. A., “Homoclinic cycles in a gene network model [in Russian],” Mat. Zamet. SVFU, 21, No. 4, 97–106 (2014).
Review
For citations:
Ayupova N.B., Golubaytnikov V.P. Phase portraits of two nonlinear models of circular gene networks. Mathematical notes of NEFU. 2023;30(2):3-13. (In Russ.) https://doi.org/10.25587/SVFU.2023.54.12.001
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