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.
Keywords:
block-linear autonomous system, phase portrait, circular gene network, fixed point of a monotonic mapping, cycle.
The authors were supported by the Russian Foundation for Basic Research (project no. 18-01-00057) and the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences No. I.1.5 (project no. 0314-2018-0011).
Citation:
V. P. Golubyatnikov, V. V. Ivanov, “Cycles in the odd-dimensional models of circular gene networks”, Sib. Zh. Ind. Mat., 21:4 (2018), 28–38; J. Appl. Industr. Math., 12:4 (2018), 648–657
This publication is cited in the following 13 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
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
V. V. Ivanov, “Prityagivayuschii predelnyi tsikl modeli nechetnomernoi koltsevoi gennoi seti”, Sib. zhurn. industr. matem., 25:3 (2022), 25–32
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. V. Ivanov, “Attracting Limit Cycle of an Odd-Dimensional Circular Gene Network Model”, J. Appl. Ind. Math., 16:3 (2022), 409
N. B. Ayupova, V. P. Golubyatnikov, “Ob odnom tsikle v pyatimernoi modeli koltsevoi gennoi seti”, Sib. zhurn. industr. matem., 24:3 (2021), 19–29
V. P. Golubyatnikov, L. S. Minushkina, “On Uniqueness and Stability of a Cycle in One Gene Network”, Sib. Electron. Math. Rep., 18 (2021), 464–473
N. B. Ayupova, V. P. Golubyatnikov, “On a Cycle in a $5$-Dimensional Circular Gene Network Model”, J. Appl. Ind. Math., 15:3 (2021), 376
V. P. Golubyatnikov, V. S. Gradov, “Non-Uniqueness of Cycles in Piecewise-Linear Models of Circular
Gene Networks”, Sib. Adv. Math., 31:1 (2021), 1
V. P. Golubyatnikov, V. S. Gradov, “O needinstvennosti tsiklov v nekotorykh kusochno-lineinykh modelyakh koltsevykh gennykh setei”, Matem. tr., 23:1 (2020), 107–122
V. A. Likhoshvai, V. P. Golubyatnikov, T. M. Khlebodarova, “Limit cycles in models of circular gene networks regulated by negative feedback loops”, BMC Bioinformatics, 21:11, SI (2020), 255
V. P. Golubyatnikov, L. S. Minushkina, “Monotonicity of the Poincaré mapping in some models of circular gene networks”, J. Appl. Industr. Math., 13:3 (2019), 472–479
V. P. Golubyatnikov, V. V. Ivanov, “Edinstvennost i ustoichivost tsikla v trekhmernykh blochno-lineinykh modelyakh koltsevykh gennykh setei”, Sib. zhurn. chist. i prikl. matem., 18:4 (2018), 19–28
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