Geodesic
Periodic solutions of travelling-wave type in circular gene networks
Izvestiya. Mathematics , Tome 80 (2016) no. 3, pp. 523-548.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider circular chains of unidirectionally coupled ordinary differential equations which are mathematical models of artificial gene networks. We study the problems of the existence and stability of special periodic solutions, the so-called travelling waves, in these chains. We establish that the number of such periodic solutions grows unboundedly as the number of links in the chain grows. However, at most one of these travelling waves can be stable. We give an explicit algorithm for choosing the stable cycle.
Keywords: chain of unidirectionally coupled equations, artificial gene network, travelling wave, asymptotics, stability. 
@article{IM2_2016_80_3_a4,
     author = {A. Yu. Kolesov and N. Kh. Rozov and V. A. Sadovnichii},
     title = {Periodic solutions of travelling-wave type in circular gene networks},
     journal = {Izvestiya. Mathematics },
     pages = {523--548},
     publisher = {mathdoc},
     volume = {80},
     number = {3},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a4/}
}
TY  - JOUR
AU  - A. Yu. Kolesov
AU  - N. Kh. Rozov
AU  - V. A. Sadovnichii
TI  - Periodic solutions of travelling-wave type in circular gene networks
JO  - Izvestiya. Mathematics 
PY  - 2016
SP  - 523
EP  - 548
VL  - 80
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a4/
LA  - en
ID  - IM2_2016_80_3_a4
ER  - 
%0 Journal Article
%A A. Yu. Kolesov
%A N. Kh. Rozov
%A V. A. Sadovnichii
%T Periodic solutions of travelling-wave type in circular gene networks
%J Izvestiya. Mathematics 
%D 2016
%P 523-548
%V 80
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a4/
%G en
%F IM2_2016_80_3_a4
A. Yu. Kolesov; N. Kh. Rozov; V. A. Sadovnichii. Periodic solutions of travelling-wave type in circular gene networks. Izvestiya. Mathematics , Tome 80 (2016) no. 3, pp. 523-548. http://geodesic.mathdoc.fr/item/IM2_2016_80_3_a4/

[1] M. B. Elowitz, S. Leibler, “A synthetic oscillatory network of transcriptional regulators”, Nature, 403:6767 (2000), 335–338 | DOI

[2] A. N. Tikhonov, “Sistemy differentsialnykh uravnenii, soderzhaschie malye parametry pri proizvodnykh”, Matem. sb., 31(73):3 (1952), 575–586 | MR | Zbl

[3] E. P. Volokitin, “O predelnykh tsiklakh v prosteishei modeli gipoteticheskoi gennoi seti”, Sib. zhurn. industr. matem., 7:3 (2004), 57–65 | MR | Zbl

[4] O. Buse, A. Kuznetsov, R. A. Pérez, “Existence of limit cycles in the repressilator equations”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19:12 (2009), 4097–4106 | DOI | MR | Zbl

[5] O. Buse, R. Pérez, A. Kuznetsov, “Dynamical properties of the repressilator model”, Phys. Rev. E (3), 81:6 (2010), 066206, 7 pp. | DOI | MR

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

[7] G. V. Demidenko, N. A. Kolchanov, V. A. Likhoshvai, Yu. G. Matushkin, S. I. Fadeev, “Mathematical modeling of regular contours of gene networks”, Comput. Math. Math. Phys., 44:12 (2004), 2166–2183 | MR | Zbl

[8] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Chaos phenomena in a circle of three unidirectionally connected oscillators”, Comput. Math. Math. Phys., 46:10 (2006), 1724–1736 | DOI | MR

[9] T. Kapitaniak, L. O. Chua, “Hyperchaotic attractors of unidirectionally-coupled Chua's circuits”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 4:2 (1994), 477–482 | DOI | MR | Zbl

[10] I. P. Mariño, V. Pérez-Muñuzuri, V. Pérez-Villar, E. Sánchez, M. A. Matías, “Interaction of chaotic rotating waves in coupled rings of chaotic cells”, Phys. D, 128:2-4 (1999), 224–235 | DOI

[11] P. Perlikowski, S. Yanchuk, M. Wolfrum, A. Stefanski, P. Mosiolek, T. Kapitaniak, “Routes to complex dynamics in a ring of unidirectionally coupled systems”, Chaos, 20:1 (2010), 013111, 10 pp. | DOI | MR

[12] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Relaxation self-oscillations in Hopfield networks with delay”, Izv. Math., 77:2 (2013), 271–312 | DOI | DOI | MR | Zbl

[13] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Periodic traveling-wave-type solutions in circular chains of unidirectionally coupled equations”, Theoret. and Math. Phys., 175:1 (2013), 499–517 | DOI | DOI | MR | Zbl

[14] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “The buffer phenomenon in ring-like chains of unidirectionally connected generators”, Izv. Math., 78:4 (2014), 708–743 | DOI | DOI | MR | Zbl

[15] A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “Relay with delay and its $C^1$-approximation”, Proc. Steklov Inst. Math., 216 (1997), 119–146 | MR | Zbl

[16] A. Yu. Kolesov, E. F. Mishchenko, N. Kh. Rozov, “A modification of Hutchinson's equation”, Comput. Math. Math. Phys., 50:12 (2010), 1990–2002 | DOI | MR | Zbl

[17] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Relaxation self-oscillations in neuron systems. I”, Differ. Equ., 47:7 (2011), 927–941 | DOI | MR | Zbl