Abstract:
We construct an invariant two-dimensional surface in the phase portrait of a certain six-dimensional dynamical system which is considered as a model for the circular gene network functioning. This invariant surface contains an equilibrium point $S_0$ of the system, and if $S_0$ is hyperbolic then this surface contains a cycle of the system.
The conditions for the existence of a cycle of this and similar systems were obtained earlier.
The author is grateful to N. B. Ayupova and V. P. Golubyatnikov for necessary
recommendations and discussions, the anonymous reviewer for the helpful remarks, and V. N.
Dyatlov for technical assistance.
This publication is cited in the following 10 articles:
Vladimir P. Golubyatnikov, Natalia B. Ayupova, Natalia E. Bondarenko, Alina V. Glubokikh, “Hidden attractors and nonlocal oscillations in gene networks models”, Russian Journal of Numerical Analysis and Mathematical Modelling, 39:2 (2024), 75
L. S. Minushkina, “Periodic Trajectories of Nonlinear Circular Gene Network Models”, Sib Math J, 65:3 (2024), 718
L. S. Minushkina, “Periodicheskie traektorii nelineinykh modelei koltsevykh gennykh setei”, Vladikavk. matem. zhurn., 25:4 (2023), 80–90
V. P. Golubyatnikov, A. A. Akinshin, N. B. Ayupova, L. S. Minushkina, “Stratifications and foliations in phase portraits of gene network models”, Vestn. VOGiS, 26:8 (2023), 758
V. P. Golubyatnikov, L. S. Minushkina, “On uniqueness of a cycle in one circular gene network model”, Siberian Math. J., 63:1 (2022), 79–86
N. B. Ayupova, V. P. Golubyatnikov, L. S. Minushkina, “Ob invariantnykh poverkhnostyakh v fazovykh portretakh modelei koltsevykh gennykh setei”, Sib. zhurn. industr. matem., 25:4 (2022), 5–13
A. F Mastryukov, “Finite Difference Schemes of 4th Order Approximation for Maxwell's Equations”, Numer. Analys. Appl., 15:3 (2022), 236
N. B. Ayupova, V. P. Golubyatnikov, L. S. Minushkina, “On Invariant Surfaces in the Phase Portraits of Models of Circular Gene Networks”, J. Appl. Ind. Math., 16:4 (2022), 589
V. P. Golubyatnikov, O. A. Podkolodnaya, N. L. Podkolodnyi, N. B. Ayupova, N. E. Kirillova, E. V. Yunosheva, “Ob usloviyakh suschestvovaniya tsiklov v dvukh bazovykh modelyakh tsirkadnogo ostsillyatora mlekopitayuschikh”, Sib. zhurn. industr. matem., 24:4 (2021), 39–53
V. P. Golubyatnikov, O. A. Podkolodnaya, N. L. Podkolodnyy, N. B. Ayupova, N. E. Kirillova, E. V. Yunosheva, “On Conditions for the Existence of Cycles in Two Models of a Circadian Oscillator of Mammals”, J. Appl. Ind. Math., 15:4 (2021), 597
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