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Volume 1, Issue 3
Limit Cycles in a Two-Species Reaction

Brigita Ferčec, Ilona Nagy, Valery G. Romanovski, Gábor Szederkényi & János Tόth

DOI: 10.12150/jnma.2019.283

J. Nonl. Mod. Anal., 1 (2019), pp. 283-300.

Published online: 2021-04

[An open-access article; the PDF is free to any online user.]

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  • Abstract

Kinetic differential equations, being nonlinear, are capable of producing many kinds of exotic phenomena. However, the existence of multistationarity, oscillation or chaos is usually proved by numerical methods. Here we investigate a relatively simple reaction among two species consisting of five reaction steps, one of the third order. Using symbolic methods we find the necessary and sufficient conditions on the parameters of the kinetic differential equation of the reaction under which a limit cycle bifurcates from the stationary point in the positive quadrant in a supercritical Hopf bifurcation. We also performed the search for partial integrals of the system and have found one such integral. Application of the methods needs computer help (Wolfram language and the Singular computer algebra system) because the symbolic calculations to carry out are too complicated to do by hand.

  • Keywords

Limit cycles, two-species reaction, third order reaction step.

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COPYRIGHT: © Global Science Press

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@Article{JNMA-1-283, author = {Ferčec , BrigitaNagy , IlonaRomanovski , Valery G.Szederkényi , Gábor and Tόth , János}, title = {Limit Cycles in a Two-Species Reaction}, journal = {Journal of Nonlinear Modeling and Analysis}, year = {2021}, volume = {1}, number = {3}, pages = {283--300}, abstract = {

Kinetic differential equations, being nonlinear, are capable of producing many kinds of exotic phenomena. However, the existence of multistationarity, oscillation or chaos is usually proved by numerical methods. Here we investigate a relatively simple reaction among two species consisting of five reaction steps, one of the third order. Using symbolic methods we find the necessary and sufficient conditions on the parameters of the kinetic differential equation of the reaction under which a limit cycle bifurcates from the stationary point in the positive quadrant in a supercritical Hopf bifurcation. We also performed the search for partial integrals of the system and have found one such integral. Application of the methods needs computer help (Wolfram language and the Singular computer algebra system) because the symbolic calculations to carry out are too complicated to do by hand.

}, issn = {2562-2862}, doi = {https://doi.org/10.12150/jnma.2019.283}, url = {http://global-sci.org/intro/article_detail/jnma/18843.html} }
TY - JOUR T1 - Limit Cycles in a Two-Species Reaction AU - Ferčec , Brigita AU - Nagy , Ilona AU - Romanovski , Valery G. AU - Szederkényi , Gábor AU - Tόth , János JO - Journal of Nonlinear Modeling and Analysis VL - 3 SP - 283 EP - 300 PY - 2021 DA - 2021/04 SN - 1 DO - http://doi.org/10.12150/jnma.2019.283 UR - https://global-sci.org/intro/article_detail/jnma/18843.html KW - Limit cycles, two-species reaction, third order reaction step. AB -

Kinetic differential equations, being nonlinear, are capable of producing many kinds of exotic phenomena. However, the existence of multistationarity, oscillation or chaos is usually proved by numerical methods. Here we investigate a relatively simple reaction among two species consisting of five reaction steps, one of the third order. Using symbolic methods we find the necessary and sufficient conditions on the parameters of the kinetic differential equation of the reaction under which a limit cycle bifurcates from the stationary point in the positive quadrant in a supercritical Hopf bifurcation. We also performed the search for partial integrals of the system and have found one such integral. Application of the methods needs computer help (Wolfram language and the Singular computer algebra system) because the symbolic calculations to carry out are too complicated to do by hand.

Brigita Ferčec, Ilona Nagy, Valery G. Romanovski, Gábor Szederkényi & János Tόth. (1970). Limit Cycles in a Two-Species Reaction. Journal of Nonlinear Modeling and Analysis. 1 (3). 283-300. doi:10.12150/jnma.2019.283
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