On the correlation between properties of one-dimensional mappings of control functions and chaos in a special type delay differential equation

V. A. Likhoshvai ; V. V. Kogai ; S. I. Fadeev ; T. M. Khlebodarova

Matematičeskaâ biologiâ i bioinformatika, Tome 12 (2017) no. 2, pp. 385-397.

Résumé

A differential equation of a special form, which contains two control functions $f$ and $g$ and one delayed argument, is analyzed. This equation has a wide application in biology for the description of dynamic processes in population, physiological, metabolic, molecular-genetic, and other applications. Specific numerical examples show the correlation between the properties of the one-dimensional mapping, which is generated by the ratio $f /g$, and the presence of chaotic dynamics for such equation. An empirical criterion is formulated that allows one to predict the presence of a chaotic potential for a given equation by the properties of the one-dimensional mapping $f /g$.

Bibliographie

[1] Likhoshvai V. A., Kogai V. V., Fadeev S. I., Khlebodarova T. M., “Alternative splicing can lead to chaos”, J. Bioinform. Comput. Biol., 13 (2015), 1540003 | DOI

[2] Khlebodarova T. M., Kogai V. V., Fadeev S. I., Likhoshvai V. A., “Chaos and hyperchaos in simple gene network with negative feedback and time delays”, J. Bioinform. Comput. Biol., 15:2 (2017), 1650042 | DOI | MR

[3] Likhoshvai V. A., Kogai V. V., Fadeev S. I., Khlebodarova T. M., “Chaos and hyperchaos in a model of ribosome autocatalytic synthesis”, Sci. Rep., 6 (2016), 38870 | DOI

[4] Kogai V. V., Khlebodarova T. M., Fadeev S. I., Likhoshvai V. A., “Slozhnaya dinamika v sistemakh alternativnogo splaisinga mRNK: matematicheskaya model”, Vychislitelnye tekhnologii, 20:1 (2015), 38–52

[5] Suzuki Y., Lu M., Ben-Jacob E., Onuchic J. N., “Periodic, Quasi-periodic and chaotic dynamics in simple gene elements with time delays”, Sci. Rep., 6 (2016), 21037 | DOI

[6] Mackey M. C., Glass L., “Oscillation and chaos in physiological control systems”, Science, 197 (1977), 287–289 | DOI

[7] Perez F. J., Malta C. P., Coutinho F. A., “Qualitative analysis of oscillations in isolated populations of flies”, J. Theor. Biol., 71:4 (1978), 505–514 | DOI | MR

[8] Ikeda K., Matsumoto K., “High-dimensional chaotic behavior in systems with time-delayed feedback”, Physica D, 29 (1987), 223–235 | DOI

[9] Bastos de Figueiredo J. C., Diambra L., Glass L., Malta C. P., “Chaos in two-looped negative feedback systems”, Phys. Rev. E Stat. Nonlin. Soft. Matter. Phys., 65 (2002), 051905 | DOI | MR

[10] Kogai V. V., Likhoshvai V. A., Fadeev S. I., Khlebodarova T. M., “Multiple scenarios of transition to chaos in the alternative splicing model”, Int. J. Bifurcat. Chaos, 27 (2017), 1730006 | DOI | MR

[11] Likhoshvai V. A., Fadeev S. I., Kogai V. V., Khlebodarova T. M., “On the chaos in gene networks”, J. Bioinform. Comput. Biol., 11:1 (2013), 1340009 | DOI

[12] Benincà E., Ballantine B., Ellner S. P., Huisman J., “Species fluctuations sustained by a cyclic succession at the edge of chaos”, Proc. Natl. Acad. Sci. USA, 112:20 (2015), 6389–6394 | DOI

[13] Varona P., Rabinovich M. I., Selverston A. I., Arshavsky Y. I., “Winnerless competition between sensory neurons generates chaos: A possible mechanism for molluscan hunting behavior”, Chaos, 12:3 (2002), 672–677 | DOI | MR

[14] Ermentrout B., Campbell J., Oster G., “A model for shell patterns based on neural activity”, Veliger, 28 (1986), 369–388

[15] Meinhardt H., The Algorithmic Beauty of Sea Shells, Third Edition, Springer, Heidelberg, 2003 | MR

[16] Meinhardt H., Klingler M., “A model for pattern-formation on the shells of molluscs”, J. Theor. Biol., 126 (1987), 63–89 | DOI | MR

[17] Philippe P., “Chaos, population biology, and epidemiology: some research implications”, Hum. Biol., 65:4 (1993), 525–546

[18] Constantino R. F., Desharnais R. A., Cushing J. M., Dennis B., “Chaotic dynamics in an insect population”, Science, 275 (1997), 389–391 | DOI

[19] Dennis B., Desharnais R. A., Cushing J. M., Henson S. M., Costantino R. F., “Estimating chaos and complex dynamics in an insect population”, Ecological Monographs, 71:2 (2001), 277–303 | DOI

[20] Dennis B., Desharnais R. A., Cushing J. M., Costantino R. F., “Transitions in population dynamics: equilibra to periodic cycles to aperiodic cycles”, J. Anim. Ecol., 66 (1997), 704–729 | DOI

[21] Maquet J., Letellier C., Aguirre L. A., “Global models from the Canadian lynx cycles as a direct evidence for chaos in real ecosystems”, J. Math. Biol., 55:1 (2007), 21–39 | DOI | MR

[22] Garfinkel A., Chen P. S., Walter D. O., Karagueuzian H. S., Kogan B., Evans S. J., Karpoukhin M., Hwang C., Uchida T., Gotoh M., Nwasokwa O., Sager P., Weiss J. N., “Quasiperiodicity and chaos in cardiac fibrillation”, J. Clin. Invest., 99:2 (1997), 305–314 | DOI

[23] Qu Z., “Chaos in the genesis and maintenance of cardiac arrhythmias”, Prog. Biophys. Mol. Biol., 105:3 (2011), 247–257 | DOI

[24] Holstein-Rathlou N. H., “Oscillations and chaos in renal blood flow control”, J. Am. Soc. Nephrol., 4:6 (1993), 1275–1287

[25] Korn H., Faure P., “Is there chaos in the brain? II. Experimental evidence and related models”, C. R. Biol., 326 (2003), 787–840 | DOI

[26] Gu H. G., Jia B., Chen G. R., “Experimental evidence of a chaotic region in a neural pacemaker”, Phys. Lett. A, 377:9 (2013), 718–720 | DOI

[27] Gu H., “Experimental observation of transition from chaotic bursting to chaotic spiking in a neural pacemaker”, Chaos, 23:2 (2013), 023126 | DOI

[28] Leloup J. C., Goldbeter A., “Chaos and birhythmicity in a model for circadian oscillations of the PER and TIM proteins in Drosophila”, J. Theor. Biol., 198:3 (1999), 445–459 | DOI

[29] Leloup J. C., Gonze D., Goldbeter A., “Limit cycle models for circadian rhythms based on transcriptional regulation in Drosophila and Neurospora”, J. Biol. Rhythms., 14:6 (1999), 433–448 | DOI

[30] Romond P. C., Rustici M., Gonze D., Goldbeter A., “Alternating oscillations and chaos in a model of two coupled biochemical oscillators driving successive phases of the cell cycle”, Ann. N.Y. Acad. Sci., 879 (1999), 180–193 | DOI

[31] Goldbeter A., Gonze D., Houart G., Leloup J. C., Halloy J., Dupont G., “From simple to complex oscillatory behavior in metabolic and genetic control networks”, Chaos, 11 (2001), 247–260 | DOI

[32] Ciliberto A., Novak B., Tyson J. J., “Mathematical model of the morphogenesis checkpoint in budding yeast”, J. Cell. Biol., 163:6 (2003), 1243–1254 | DOI

[33] Gérard C., Goldbeter A., “From simple to complex patterns of oscillatory behavior in a model for the mammalian cell cycle containing multiple oscillatory circuits”, Chaos, 20:4 (2010), 045109 | DOI

[34] Zhang Z., Ye W., Qian Y., Zheng Z., Huang X., Hu G., “Chaotic motifs in gene regulatory networks”, PLoS One, 7:7 (2012), e39355 | DOI

[35] El'sgol'ts L. E., Norkin S. B., Introduction to the theory of differential equations with deviating argument, Nauka, M., 1971 | MR

[36] Likhoshvai V., Ratushny A., “Generalized hill function method for modeling molecular processes”, J. Bioinform. Comput. Biol., 5:2B (2007), 521–231 | DOI | MR

[37] Feigenbaum M. J., “Universal Behavior in Nonlinear Systems”, Los Alamos Science, 1:1 (1980), 4–27 | MR

[38] Feigenbaum M. J., “The Universal Metric Properties of Nonlinear Transformations”, J. Stat. Phys., 21 (1979), 669–706 | DOI | MR

[39] Li T. Y., Yorke J. A., “Period three implies chaos”, Amer. Math. Monthly, 82:10 (1975), 985–992 | DOI | MR

[40] Ilyashenko Yu. S., “Teoremy konechnosti dlya predelnykh tsiklov”, Uspekhi Matem. Nauk, 45:2(272) (1990), 143–200

[41] Ilyashenko Yu. S., “Teoremy konechnosti dlya predelnykh tsiklov: skhema obnovlennogo dokazatelstva”, Izv. RAN. Ser. matem., 80:1 (2016), 55–118 | DOI

[42] Écalle J., Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualites mathematiques, Hermann, Paris, 1992 | MR